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This seems to be a problem that is slowly losing importance since at least in the international airport in Tel Aviv, Israeli authorities are no longer stamping passports for tourists at their arrival or departures.
Still there are people who carry old stamps in their passport or who have used one of the land border crossings where passports are still being stamped.
On the other hand it is important to note that the most common information around is that those with a Israeli visa, an Israeli stamp, an Israeli passport or a stamp of Egypt or Jordan from the border crossings with Israel are not allowed to enter Qatar.
What it happens is that immigration authorities at the airport in Doha do not really inspect the passport looking for such things overall, but there have been scarce online reports of a handful of persons who have been denied entering Qatar because of those kinds of stamps.
The best thing at this point is to be prudent, if one has an Israeli stamp in the passport and it is very noticeable probably it is better to contact immigration authorities from the country of ones origin and explain the situation, most times they are willing to extend a new passport to prevent problems.
On the other hand a little planning is a always good. Having a stamp of Qatar in the passport will not automatically exclude one of entering Israel, it is true that some people with such kind of stamps have experienced a very hard time at immigration on arrival or departure in Israel, but we have not heard of anyone denied entrance just because of the stamp.