Well so far caffeine doesn't make Methotrexate less effective:
http://www.ncbi.nlm.nih.gov/pubmed/16821266 Well according to that anyway but there's another study that suggests it does make MTX less effective and drugs.com sides with the study that suggests caffeine makes MTX less effective.
http://www.drugs.com/food-interactions/methotrexate.html?professional=1
Yet that study is talking about high amounts of caffeine intake per day (180mg+) yet the study isn't about going caffeine free, just to not have too much. This site shows the amount of caffeine in each caffeinated beverage:
http://www.runtricities.org/html/WebArticles/Caffeine.htm As you can see tea has a lower content than coffee so you could get away with more than one cup a day. There are many different teas out there that don't contain any caffeine though if you were really concerned.
Even though drugs.com states, "It may be appropriate to limit or avoid caffeine intake if you are taking methotrexate. Avoid drinks, foods, or diet pills that contain caffeine, such as coffee, tea, cola, and chocolate. Taking a stimulant together with methotrexate can increase your risk of unpleasant side effects This can cause headache, lightheadedness, restlessness, or insomnia." They provide no proof to back up this claim. All they do is show the study I provided above which made MTX less effective in excessive coffee drinkers and the study states that that is why the patients stopped taking MTX. The study said nothing about an increase in side effects. If anything they're just listing possible side effects from caffeine in general.
"Thus, caffeine (contained in coffee), an adenosine receptor antagonist, appears partially to block the effects of methotrexate in patients with RA."
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1798096/ This is providing possible proof that caffeine may make MTX less effective.
Now the studies may be low dose vs high doses of MTX but the one that states that MTX does not affect the effectiveness of MTX is more recent than the one that suggests caffeine may make MTX less effective. Personally I tend to side with more recent studies. They are both published the same year but one is in April and the other is done in July.
Everything in moderation is important though.