The bonus of cash or gift will be reported to the IRS under either 1099-INT or 1099-MISC. You are likely to have to pay taxes on the bonus. If the bonus is a piece of merchandise, because they will report the fair market value of the item.
If we are talking about bank accounts, there isn't a risk of your credit score being impacted.On the other hand, opening credit cards for the bonus does have some impact on your credit score.
There is a risk if your employer requires you to report your financial info on a yearly basis: the number of bank accounts could make them concerned. Large number of these types of accounts could complicate your application for a mortgage.
There are opportunity cost issues. For example: you don't want to pull that $5,000 out of the bank to fix the transmission because it will cost you the $100 bonus; if only we can limp through the next two weeks we will be fine.
Assuming you aren't making large cash deposits, the IRS won't be concerned about the transactions.
If the bank requires direct deposit, then asking HR to adjust the direct deposit forms could make the person responsible for doing it very grumpy.