Although PSP titles are getting the HD treatment like its PS2 brethren, the remastered games will not be getting any Trophy support, despite support for a myriad of other things, including stereoscopic 3D and shared data.
The feature, or lack thereof, was confirmed by Sony when asked if Tophies were to be included in the Remasters brand. Unfortunately, it will not.
Siliconera speculates that one of the reasons for the exclusion is because of how widely cracked open the PSP is. It's easily the most hacked handheld device alongside the Nintendo DS. Since Remaster titles will be sharing data between PS3s and PSPs, it is quite possible for hackers to find an exploit to obtain Trophies through cheating.