Nope. He does serve for a ten year term, but he serves “at the pleasure of the president”.
It’s unusual to fire the FBI director, but it’s been done and if anyone deserves the boot it’s James Comey.
I stand corrected, Read This:
28 U.S.C. section 532 provides that the Attorney General may appoint the Director of the FBI. No limitation is placed on the ability of the AG to fire the FBI director. As such, the AG may fire the FBI director at will.
If the President wanted to, the President could fire the FBI director by asking the AG to do so. Congress need not approve.