There shouldn't be tax implications unless you withdraw money from the account. Even then, you likely won't pay any taxes on a withdrawal if the money is spent on qualified medical expenses.
Fees are dependent on the individual account.
I also have an HSA within a TD Ameritrade account. In my situation I have not encountered any extra overhead for dealing in mutual funds. However, I had to explicitly sign up for Commission-Free ETFs to avoid extra fees when dealing in ETFs, which still only impacts a subset of ETFs. They will change me standard trading fees for dealing with stocks.
I wouldn't want to venture any assumptions about the fees involved in your account because they could easily be impacted by things like how much money you are holding in the account. I have seen situations in the past where an account would encounter fees based on the number of mutual funds being held as well.
You should inspect your own Fee Schedule and determine other ways to avoid fees, such as using electronic delivery of documents.