With the information you have given, I would say never. Remember the banker is a salesman, and the line of credit is the product. If you don't need to borrow the money for something specific, then you don't need the line of credit in the first place. Even if you did need something I would tell you to save up and pay cash for it.
On the tax advantage: There is none, in the US you can deduct your mortgage interest on your taxes but it's not a tax credit it's a tax deductions. Let me explain further:
You spend $10,000 on mortgage interest, and you're in the 25% tax bracket.
You send the bank $10,000 in return you get at tax savings of $2500.
You are still in the hole $7500
You would have been better off not taking out the loan in the first place.
On the Emergency Fund: You should have 3 - 6 months of expenses in cash, like a money market account.
This money isn't for investing, it's like insurance, and you don't make money on insurance. The last thing you want to do is have to go into debt right in the middle of an emergency. Say you lost your job, the last thing you would want to do is borrow money, right at the time you have no income to pay it back. The bank is under no obligation to maintain you credit limit and can without notice reduce it, they can in most cases call the loan balance due in full with little or no notice as well. Both of those are likely scenarios if the bank were to become aware of the fact that you were unemployed.