Everything I've ever read says while the once used the FN/FAL, they now use either the Galil or the M-16. The Galil came/comes in both 7.62 and 5.56, but the 5.56 is more common, except in specialize applications. Every photo I've seen of IDF troops show M-16s and the occasional Galil. The woman troops, and military and civilian police, do carry the UZI. Here's what Wikipedia has to say about Israeli use of the FN/FAL:
The FN FAL saw action in the Suez War (1956), Six Day War (1967) and Yom Kippur War (1973) but received unfavorable reviews by the soldiers who use it. Its main shortage was the inability to carry out full automatic fire and its lack of durability to harsh desert condition. The rifle performed poorly in sandy environment as sand or mud cause it to jam rapidly. The rifle was being replaced during the later stages of the Yom Kippur War when Israeli soldiers replaced for American emergency aid M-16 rifles, and AK-47 assault rifles that were taken from dead and captured Arab soldiers. The Israeli IMI Galil also saw limited action and prove itself highly durable and reliable in the harsh desert conditions of the Sinai and the muddy Golan Heights.
The FN FAL were officially abandoned in 1975, and were replaced by the IMI Galil as the IDF standard issued weapon.
Wikipedia also has articles on the Galil
It's heavy but being somewhat of an AK derivative, reliable. It's also expensive to produce, which is why the Israelis use lots of M-16 variants.
http://www.thehighroad.org/showthread.php?t=170763
(...including this: "Without a single exception, functional problems with the FAL were, in [the Israeli military contact's] opinion, the result of either substandard or non-existent maintenance on the part of Israeli personnel. In other words, the gun worked fine as long as routine maintenance was performed.....")
(Sounds just a bit like the M-16, doesn't it? ;>)
For a more applicable (and better documented) reference, just check out "Ol' Dirty - A Texas Legend:"
http://www.falfiles.com/forums/showthread.php?threadid=68486&perpage=30&pagenumber=1
As per a recent update, "Ol' Dirty" (a home-built West German G-1 FAL) had 10,000 rounds (plus 'swamp tosses' ;>) under it's belt - without cleaning - and was still running strong. Try that with your favorite AR...
(Speaking of which - suggesting that the Israelis replaced the FAL with the M-16 due to undocumented reliability problems would be akin to suggesting that the US replaced the M-14 with the M-16 because the M-14 was unreliable... ;>)