Animagi are the spirit animal of a magical person. So what if a person's personality changes, will their Animagus form change as well, as Nymphadora Tonks' Patronus did? Or will it remain?
A person's Patronus form can change, if the caster had undergone a fundamental change in worldview. But their Animagus form is Carved in Stone.
The Animagus form seems to be (although there is very little known about it) to be determined at the time the caster first masters the transformation. The form is an expression of their personality at that time. So if a person changes, their Animagus form remains constant. For this reason some people "abandon" their animagus form, as they do not like it, instead of working to alter the form.
The Patronus form seems to be (again, very little data in canon, other than Tonks and possibly Snape changing Patronus form) to be based on their worldview/spiritual outlook at the time of casting a specific Patronus.
Nymphadora Tonks was a metamorphmagus, Unlike animagi, metamorphmagus is born with this ability, while animagus has to "learn" to change. therefore Animagi can change into only one animal, while metamorphmagus can change their appearance anyway they like, not neceserilly into animals. In deathly hallows we can see her change her facial features at will, something no animagi could do. "Metamorphmagus are born, not made." - These are her words. But if you're reffering to the patronus then - Yes, patronus form can change - mainly to reseble the one of our loved ones. Tonk's patronus changed from a hare to a wolf to resemble Lupin's. Snape's patronus was the proof for Dumbledore, that he still loved lily - it was a doe, just like hers.