1996 inside the dome. Between 1999-2029 outside the dome (and probably closer to 1999 than 2029)
All three newspapers shown in the show are dated for 1996. Since we know that Truman is literate/numerate and takes the newspaper daily, this must be the year inside the dome.
Thursday, December 12th, 1996
The third reads Friday, December 13 and is copyrighted for 1996.
The date on the cover of the 'Dog Fancy' magazine reads "Novmbr 96"
And Truman's copy of SHE magazine is dated "...R, 1996"
The date on a magazine shown outside of the show reads 1999. Since the show is in its 30th year, this gives us an upper bound of 2029 for the show to take place, given that we don't know whether the magazine was published in the show's first year or its current year.
We also see a lunch menu in a scene that takes place outside the dome. It reads "3- 27 - 97".
While it seems plausible that the menu has been stuck on the wall directly under the TV for a couple of years, it seems much less likely that it would be still there after 30 years, which suggests that the magazine date is roughly current.