Why doesn't the Israeli football team play in The Mideast tournament?

[TAT]

or at least in Asia cup matches

[DanE]

Clubs who do well in either the Liga Al or State Cup qualify to compete in various UEFA-organised Europe-wide competitions in the following season (as well as continuing to play in domestic competitions). The number of Israeli clubs playing in Europe in any one season can range from four to six, depending on the qualification scenarios.

In addition, once in a European competition, it becomes possible to qualify for others:

All the winners of the Champions League Third Qualifying Round go forward to the Champions League
All the losers of the Champions League Third Qualifying Round go forward to the UEFA Cup
All the winners of the UEFA Cup Second Qualifying Round go forward to the UEFA Cup
All the winners in the Intertoto Cup Final Round go forward to the UEFA Cup Second Qualifying Round
Any clubs playing in the Champions League that finish third in the group stage go into the UEFA Cup Third Round

The Israel National Football Team is the national football team of Israel and is controlled by the Israel Football Association.

Israel has participated in five different continents: Asia, Africa, South America, Europe and Oceania in the Football World Cup qualifiers, before settling in Europe as a member of UEFA in 1994. The Israel Football Association (IFA) was founded under the name Palestine Football Association in 1928, 20 years before the establishment of the state of Israel during the Mandate for Palestine. PFA was accepted as a full member of FIFA in 1929. PFA first dealt with local leagues, but later also managed the national team that was created to participate in the World Cup qualifications (participated in 1934 and 1938 as Palestine/Eretz Yisrael national team).

The Israeli side has only appeared in one World Cup (1970).

[MichaelJ]

The simple answer is that the surrounding Arab countries have refused to play against Israel, forcing Israel to compete in Europe. Also, it wouldn't be safe to send Israelis to many countries there for matches.