Unless it is a Title 2 firearm, everything can go except the serial number. Your state or local laws may differ.
For Title 1 (non-NFA firearms), the following is unlawful:18 USC Sec. 922 It shall be unlawful for any person knowingly to transport,
ship, or receive, in interstate or foreign commerce, any firearm
which has had the importer's or manufacturer's serial number
removed, obliterated, or altered or to possess or receive any
firearm which has had the importer's or manufacturer's serial
number removed, obliterated, or altered and has, at any time, been
shipped or transported in interstate or foreign commerce.
For Title 2 (NFA) firearms:26 USC Sec. 5861 Identification of firearms other than destructive devices
Each manufacturer and importer and anyone making a firearm shall
identify each firearm, other than a destructive device,
manufactured, imported, or made by a serial number which may not be
readily removed, obliterated, or altered, the name of the
manufacturer, importer, or maker, and such other identification as
the Secretary may by regulations prescribe.
It shall be unlawful for any person -
(g) to obliterate, remove, change, or alter the serial number
or other identification of a firearm required by this chapter; or
(h) to receive or possess a firearm having the serial number or
other identification required by this chapter obliterated,
removed, changed, or altered; or
(i) to receive or possess a firearm which is not identified by
a serial number as required by this chapter
Other info:Non-NFA marking requirementsNFA marking requirementsFor non-NFA firearms, it is unlawful to remove the serial number. No prohibition is made on the removal of other required markings. For NFA firearms, not only is the removal of the serial number prohibited, but also the removal of other required markings.