Why don't we pronounce the second ש in יששכר?
Minchas Shai (to Gen. 30:18, the first place where the name appears) cites Radak, who says that it this is an example of elision: the sound of the second letter is combined into that of the first. As another example, he gives מחצצרים (I Chron. 15:24 and in a few other places in Chronicles), where the second צ is silent.
That said, as Yahu noted, there are various customs according to which the second ש in יששכר is pronounced at least some of the time. Indeed, one such custom is based on the idea that Yissachar "donated" the second ש of his name to his son יוב (Gen. 46:13), who is later called ישוב (Num. 26:24) - and that therefore יששכר should be pronounced with both ש's up to the latter point. A summary of the whole issue is here.
Rabbi Bogomilsky compiles several opinions about how to say the name, including the ones mentioned in other answers.
In short (see there for sources):
Read with two "ש" until Bamidbar 26:24, then it is read with only one "ש". As Alex answered.
Read with two "ש" the first time, after that it is only read with one. As Yahu answered.
Chabad always reads it with one, for the reason given by Gershon above.
Some have the custom to always read it with two. They bring a hint from Psalms 63:13 where it says "כִּי יִסָּכֵר פִּי דֽוֹבְרֵי־שָֽׁקֶר", which they says hints to us that who ever reads it with one "ש" is saying falsehood.
Rabbi Menachem Kasher brings most of these opinions (and others) and discusses them in his Torah Sheleimah Parshat Vayeitzei, footnote 64.
Listen to a short lecture by Rabbi Paltiel here. He discusses Rabbi Menachem Kasher's discussion.
Why is there a silent ש in the name Yissochor? The Daas Zkeineem tells us the reason Layoh put two letters of ש in t Yissochor: One for the שכר-reward for giving Zilpoh to Yaakov as a wife, the second for giving the flowers that were to help Rochail become pregnant. To reveal the story of the flowers, however, would possibly embarrass Rochail. In order not to shame her in public, Layoh made the second letter silent. — The Parasha Guide - Yeshivas Chofetz Chaim Talmudical Academy of Baltimore.
According to Rabbi Jeremy Weider (in his Introduction to Bible course, which can be found on YUTorah.org) it is a kri u'ketiv. Because this is a common kri u'ketiv it is written in the text without note (like the spelling of naarah without the final hey and other examples).
Yissachar in general represents the study of Torah. Many of the members of the Sanhedrin, the Jewish supreme court, were of the tribe of Yissaschar. It is said that Zebulun brought in the cash and supported Yissachar in their study. It was a partnership of sorts.
Now the Torah has two parts to it: the revealed and the hidden, or the legalistic and the Kabbalistic. The two s’s in Yissachar’s name correspond to these two levels in Torah, the silent “s” corresponding of course to the “silent” hidden part of Torah (Rabbi Levi Yitzchak Schneerson).
It is brought down from the Chasam Sofer (I do not remember where I saw it) that the first time יששכר occurs (when he was named) should be read pronouncing both sins. This does not really answer your question but it may provide a clue towards the answer.