Inside Star Trek: The Real Story quotes the third season budget at $178,500 per episode (page 399). The second season budget was $187,500 per episode (page 317) according to the same book. Memory Alpha quotes the average first season budget at $190,635 per episode, although skimming I can't find the page number.
Although the decrease might not seem like a lot of money, when you take into account two years of salary increases for each of the regular and semi-regular actors (not to mention the additional increase in pay Nimoy negotiated for after the first season) it is. It doesn't help that many of the costs of production are fixed. Since you can't cut the salaries being drawn for above-the-line costs (writing, directing, acting) -- many of which are actually increasing -- that means the money has to come from below-the-line areas of the production. That's one of the reasons why only two episodes in the entire season went on location.
I figured performer salaries were an issue and a reason why the budget was an issue in the third season. However, given only three of the cast members received top billing, I'm not sure how much of an issue this was in reality.
Either way, I didn't get the impression that the last season was substantially cheaper looking than the others. The writing took a nosedive. I think "That Which Survives", or "This Way to Eden", "Spock's Brain", "And the Children Shall Lead" and "Turnabout Intruder" prove this. There were still a number of good, classic episodes regardless.
One thing I have noticed about season 3 is that the costume budget for female guest stars was much, much skimpier than in previous seasons!