I have an account with Bank of America who removed some features of the account without my consent and I want to get back those features for the same price it cost when I opened it. I guess that persuading or yelling at staff them would not make any difference (more likely do even worse), but if I tell that I will otherwise close the account (thus they will lose a lot of money from my savings), would they stand back? Does the manager have any pressure to keep the account to the point of giving away free products to keep the costumer or do they not really care?
If this matters to you a lot, I agree you should leave. My primary bank account raised chequing account and transaction fees. I left. When I was closing my account the teller asked for the reason (they needed to fill out a form) and I explained it was the monthly fees. Eventually, if a bank gets enough of these, they will change.
I want to get back those features for the same price it cost when I opened it
They are in their rights to cancel features or raise prices. Just as you are in your rights to withdraw if they don't give you a deal. The reason why I mention this is that this approach is comical in some instances. A grocery store may raise the price of carrots. Typically you either deal with it or change stores. Prices rise occasionally.
thus they will lose a lot of money from my savings
From my understanding, a bank makes a large chunk of their money from fees. Very little is from the floating kitty they can have because of your savings. If you have an investment account with your bank (not recommended) or your mortgage, that would matter more. I've had friends who have left banks (and moved their mortgages) because of the bank not giving them a better rate.
Does the manager have any pressure into keeping the account to the point of giving away free products to keep the costumer or they don't really care?
Depends. I've probably say no. One data point is an anecdote; it is expected in a client base of thousands that a few will leave for seemingly random reasons. Only if mass amounts of clients leave or complain will the manager or company care.
A note: some banks waive monthly account or service fees if you keep a minimal account balance. I have one friend who keeps X thousand in his bank account to save the account fee; he budgets a month ahead of time and savings account rates are 0% so this costs him nothing.
Take your business elsewhere, where the products and services are priced at a level you agree to pay. This does two things. First, you end a bad business relationship. Why bad? Because you're not happy with the deal. Second, it sends an unambiguous signal to the losing bank that you were unhappy with their service. If they offer an exit survey, complete it, and be sure to tell them what made you unhappy with their service.
In a free market economy, if consumers all take their business where the terms are favorable, supply and demand would force the banks to compete for consumers' business.
To avoid going on and on in the comments I'm going to add this point that seems to be missing from the other answers.
"Banks often offer me deals while negotiating to open an account (since they are under high pressure to open an account)"
Would these happen to be the regionally advertised account opening deals like a $200 new checking account bonus if you deposit at least $x and leave it for at least 90 days? This kind of deal is not unique to you. This is not offered to you because of your unique negotiating ability.
You need to understand the authority of the person you're dealing with. Products are designed in the corporate arm of the bank. Once a product is ready, it's rolled out to branches to be sold; sometimes with some fancy sign-up bonus. A checking account is a product, just like an iPhone. Apple took the headphone jack out of the iPhone 7, no amount of negotiating with the Genius at the Apple store will put it back for you.
Vote with your wallet, show the bank you're unhappy by leaving.
I would hold off on making that threat (closing your account). First, because as others have said, it's not likely to help. And second, assuming you're willing to make good on that threat, you should only play that card as a final absolute last resort, because if it fails, and you close your account, there is little to nothing else you can try to get what you want.
First, talk one-on-one with a personal banker at your local BA branch. You might be surprised at how helpful they can be.
Next, try talking to customer service on the phone. After that, you might try sending a letter to corporate HQ.
A lot depends on the particular "feature" you are talking about and why they removed it. It could be that 1) the bank finds the feature is just too costly provide for free, 2) there may be a technical reason why they can no longer provide it, 3) it could be as simple as that few to none of their customers (excluding you) are actually using the feature, or 4) it could be that due to changing regulation, or market forces, no bank is offering that feature anymore.
Also, while they may not care specifically about your business, the local branch has an incentive to not drive customers away if it can be reasonably avoided.
The specific answer to your question
"Does threatening to close the account ... [give me negotiating power]"
The simple factual answer is:
1) For large banks, eg BoA in your example, simply no, not at all. (*)
2) In the US milieu, for local credit unions, yes. The manager or any senior staff member - really any staff member at your local branch is under high pressure to keep accounts and add new accounts. if you are polite but genuinely and clearly (politely) express that you "may well have to close your account over the issue", you can often get fees refunded or whatever.
(*) Footnote, the big banks are so useless - there isn't even anyone you could explain your "demand" to. The min-wage employees you're talking to would just glaze over and look through you, it would mean zero.
From the bank's perspective, they are offering a service and within their rights to charge appropriately for that service. Depending on the size of their operation, they may have considerable overhead costs that they need to recoup one way or another to continue operating (profitably, they hope).
Traditionally, banks would encourage you to save with them by offering interest growth on your deposits. Meanwhile they would invest your (and all of their customer's) funds in securities or loans to other patrons that they anticipate will generate income for them at a faster rate than the interest they pay back to you. These days however, this overly simplified model is relatively insignificant in consumer banking. Instead, they've found they can make a lot more profit by simply charging fees for the handling of your funds, and when they want to loan money to consumers they just borrow from a central bank.
What this means is that the size of your balance (unless abnormally huge) is of little interest to a branch manager - it doesn't generate revenue for them much faster than a tiny balance with the same number of transactions would. To put it simply, they can live without you, and your threatening to leave, even if you follow through, is barely going to do anything to their bottom line. They will let you.
If you DO have an abnormally huge balance, and it's all in a simple checking or savings account, then it might make them pause for thought. But if that's true then frankly you're doing banking wrong and should move those funds somewhere where they can work harder for you in terms of growth. They might even suggest so themselves and direct you to one of their own "personal wealth managers".
This is one of those questions where if you have to ask then the answer is no, they will not negotiate with you. Pretty much the only chance you have to get special treatment is if you have a private banker assigned to your portfolio of accounts.
I am sure they do not want to lose you as a customer because even if you hold enough in balances to get your fee waived they are still making money on your deposit. However, what little they make from you is not enough to carve out special features for your accounts. If you are the type of customer that has this kind of sway you likely already know it.