TLDR: There are no to few monetary downsides.
The process of settling an estate is called probate. Creditors can make claims against the estate, and assets should stand to pay any debts. If more debts are owed then assets, the beneficiaries are not held liable. Final expenses are usually the first amount paid out in full. So if the estate only contains enough assets to pay final expenses, then the creditors receive nothing. Usually creditors are paid pro-rated if there is not enough to cover debts.
For the record, the probate process is greatly simplified if one has a will. Get a will if you don't have one.
Life insurance is a bit different though. It passes directly to the beneficiaries and depending on the state could be untouchable by creditors.
The same thing could also happen with retirements accounts.
With 401K accounts, you could take some of it out, and pay tax on that. You could also roll it into your own account.
Property receives a really good benefit. While it does pass through probate the cost basis of real estate is reestablished at the time of death. So if grandpa bought a house for 30K in the 40's, and it is now worth 120K. You inherit a 120K piece of property and when you sell you use 120K as your cost basis not 30K.
Any estate taxes are typically paid by the state, not technically the heirs. If there is a 5% estate tax and they are to inherit 100K, they will only receive 95K. They will not receive 100K then be expected to be paid 5K. "Electrically" the same, but a large difference in responsibility.
The biggest downside is if you have a will fight on your hands. If someone disputes the validity of the will that can incur a lot of legal fees. For small estates, it may not be worth the fight.
The next is if any assets do go through probate. The process is lengthy and depending on the executor, they could reduce the size of the estate for charging for their time.