What are the effects of getting lidocaine injected in your arm for a pain free tattoo removal?

No problem. It will numb the skin and will wear off in an hour or so. As long as you do not exceed recommended dosage there is no long term effect.
Better ways. To do this. There are Lidocaine creams/ointments that can be applied that will work just as well and no need to distort the tissue with injection. The effects are that if too much is injected or it is injected into a blood vessel you can have a severe overdose. Its a medicine and should only be handled by those who have training.