Obviously, out of universe having these alien species would be a huge headache for make-up and financial drain in terms of production costs, but is there an in-universe explanation? Tellarites and Andorians are founding members of the Federation, so it surprises me we don't see them more often. I know Vulcans were founding members as well, but their relative absence has been explained.
4 Answers
Andorians
Not meaning to take Nate Watson's point, but they suffered a serious population problem; they had great difficulty in replenishing the species:
From an initial population of 3 billion, there were only 90 million Andorians by 2376 (DS9 novel: Andor: Paradigm)
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Tellarites
Considering that the majority of Starfleet Vessels we see are human-dominated, it may be due to climate:
Tellarites found Human room temperature to be cold, indicative of a higher body temperature. (ENT: "Babel One")
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Both
It seems that another reason may be mutual distrust of one another:
Despite years of co-existence with the Tellarites, the Andorians were known to hold hostile relations with this founding Federation member race.
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This is supported by Ambassador Gral, a Tellarite:
Ambassador Gral recalled being aboard a Tellarite cruiser as they drove "those blue demons" back into their own territory by force. (ENT: "Babel One")
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Andorian Defense Force was very much in service in 2375. It was part of the 5th Fleet which was a standing fleet, tasked with defending the Core worlds in the Andorian Sector.
The Tellarite no longer operated a military after the founding of the Federation. The Tellarites are usually behind the famed engineering in the Federation, as cited by the Dominion, "The famed starfleet engineers that can turn rocks into replicators", was actually directed at the Tellarites.
Vulcans maintained a Science Fleet, mostly working spiritual beliefs surrounding their logic philosophy.
The ships that we see have human centric populations as I imagine it is easier to have ships with races very closely matched environmentally.
In the DS9 baseball episode (Take me out to the holosuite) we see that the vulcan captain has a (primarily) vulcan crew
In the Andorian's case, the reason is biology. As of the time of the Enterprise relaunch novels, the Andorian population was declining, partially due to their reproductive process. Because of this, their population was so low, they'd naturally be rare on starships.
As an additional effect, this probably meant that Andorians were less likely to join Starfleet, because of the high redshirt death rate. Why join an organization with a high death rate, when your people are in a population decline?
Regarding the Tellarites, I don't have much of an answer for them. The closest answer I can come up with is that Tellarites are short. You're less likely to see them, simply because you see over them. It's not perfect, but it's an answer.