I understand that Qatar is a slightly odd example as you require an exit visa to leave the country (although I understand that this is under review by the state) and your employer has to co-operate. (See the Middle East forum.) There have been cases of staff resigning by e-mail whilst abroad (but make sure that you don't return to the country).
In the Western world there is absolutely nothing to stop you leaving your company (its may be different for a military pilot) but don't expect a glowing reference if you don't serve your notice period.
Some companies try and enforce non compete clauses for managers (I don't know about pilots) but UK courts tend not to be too strict in enforcing them viewing them as restricting competition.
Pilots careers are based on seniority (the longer you have served the better the pay, choice of types, schedules, etc) which is generally not transferrable between airlines and this may well be a consideration for a pilot who has been with a legacy carrier for a long time (not much of an issue if you fly for certain US commuter carriers).