Martin pours olive oil on top of the waiter's head, and says that in 1925 slicked back hair was going to become all the rage. In fact, slicked back hair with brilliantine was already well in vogue, as can be seen in this Rudolph Valentino film, Eyes of Youth (1919). It might have even started even earlier than that.
A little before the 20 minute mark, Martin says to Tim, "We're not expecting our next solar eclipse until March 1932." This scene takes place in 1925. While there were a number of solar eclipses between 1925 and 1932, if we limit our search to ones visible from Hollywood (where this episode takes place), the next solar eclipse would have been April 28, 1930.
When Tim and Martin watch a film made in 1925, it is shown in color on television in 1965.