The Litvish/Russian pronunciation of the Cholam sound is an /ay/ sound, similar (I think actually exactly the same) as a tzeirei. Are there any sources saying that it is a Halachic problem pronouncing the cholam like a tzeirei?
It's generally accepted that traditional pronunciation schemes for Hebrew are always halachically acceptable. The basic mekor for this is that the Gemara (e.g. Megillah 24b, brought down by the Shulchan Aruch) implies that there are halachic problems with pronouncing ayin as aleph. Numerous mefarshim qualify that this does not apply in a place where the common pronunciation merges these sounds; the problem is only if this is an uncommon or distracting pronunciation in the locale where it is being used (see the mefarshim on the Shulchan Aruch which brings the previous Gemara down, c.f. Igros Moshe OC 3:5).
While the sources generally refer to pronunciation differences in general, there are a few specific references to the Litvish pronunciation. Aruch Hashulchan OC 62:2 specifically states that it is acceptable that many Litvaks pronounce cholam as /ej/, and both sin and shin as /s/ (the infamous "Sabosdiker losn"), although he mentions that a person who is careful to pronounce the words "avodah" and "aveidah" differently will be rewarded greatly (since the word "aveidah" has highly negative connotations). I also remember a source poking fun at Litvaks for pronouncing "kavod va'oz" as "kaveid va'eiz" ("liver and goat"), although at the moment I can't seem to find it.