Races in Israel late February?

As far as marathons are concerned, you're out of luck. You'll be here between all of them. See marathon calendar at:
http://marathons.ahotu.com/calendar/marathon/israel

If you want more comprehensive information, perhaps contacting the Tel Aviv Running Club would help. Their web site is at:
http://marta.org.il/english.htm

You're arriving just after a great Dead Sea Half Marathon, and leaving right before the Tel Aviv and Jerusalem Marathons...

Anyways, there is some stuff going on -

Feb 24-26 - Eilat - Tour Eilat Bike Race which includes
10k dirt road race and various distances in road races (70k, 136k)

Feb 25 - Even Yehuda Race which is a running race with 10k and 5k heats
Even Yehuda is a town about 35 minutes northeast of Tel Aviv and probably a major pain in the ass to get to by public transport

Feg 25 - Begin Race - Jerusalem - it's a preparation race for the Jerusalem Marathon, with 8 and 16km races. This race is free, but you still need to sign up.
This is the poster with all the info in English at the bottom - http://www.realtiming.co.il/site/271
This is the registration in Hebrew and Englsh - http://www.realtiming.co.il/site/271/page/85

March 3 - Kfar Saba Race in Kfar Saba, about 20 minutes from Tel Aviv and probably a minor pain in the ass to get to by public transport. This is a 10k running race which will probably be the most commercial of all the above, with over 1000 runners. It kicks off at 9:45 am, until Feb 26 it's 90 NIS to sign up, after that it's 110 NIS. It's with a chip, shirt and all that stuff...

Jerusalem will probably be the easiest for you, as you probably plan on being in Jlem anyhow and its free.

Kfar Saba will likely be the most "racey" race, with official timing and an actual organizing committee production and all that.

Let me know if you need help signing up or anything

The fact that Even Yehuda is phenomenally easy to reach by public transport, being within walking distance of Beit Yehoshua railway station, is irrelevant because February 25 is a Saturday, when public transport doesn't run. Same applies to the Kfar Saba event on March 3.

Another vote for the Jerusalem race!

Good point about Saturday - but The Beit Yehoshua train station is a good 2 km from the start point as the crow flies, probably 3-4 if you're taking the paved road.... though it could be a good warmup ahead of the race...

Great suggestions! Thanks everyone for the fast responses. I'll probably take a couple of days to look the info over and get back to you with other questions. Does anyone know about races in the West Bank?

Sorry to ask more, but it looks like the races that were mentioned above won't fit in my itinerary. Do you have any suggestions for races in the North 21 February to the end of the month?

Races are only held on the weekends - which gives you only one weekend - of February 24-25.
The only races on that weekend are
Tour Eilat - a bike race in Eilat on the 23rd
Begin Race in Jerusalem on the 24rd
Even Yehuda 10k Race on the 25th
Ramat haSharon on the 25th

There is also an Orienteering race on the 24th in north Tel Aviv...