Lift, raise, rear, elevate, hoist, heave, boost are comparable when meaning to move from a lower to a higher place or position.
Lift often carries an implication of effort exerted to overcome the resistance of weight, but it may be extended to whatever rises high by natural or artificial means or processes or to something immaterial that rises or is made to rise (as in spirit, in feeling, or in aspiration).
Raise may suggest less effort than lift, but it carries a stronger implication of bringing something to the vertical or to a high position for which it is fitted by nature or intended function; thus, one raises a pole by setting it on end, but one lifts it by picking it up; a flag is raised to the top of its staff, but it is lifted when held high enough to be seen. In extended use raise may imply a lifting to a higher level (as of worth, efficiency, or accomplishment).
Rear is often used in place of raise, but, unlike raise, it can be used intransitively with the meaning to raise itself or, in the case of a horse, to raise its forelegs.
Elevate may be used in place of lift or raise in certain collocations where it does not seem unduly formal or pretentious, but, in general, the word suggests exaltation, uplifting, or enhancing.
Hoist implies raising something heavy aloft, often by such mechanical means as a tackle.
Heave implies a lifting upward or onward with strain or effort usually by impulsion from without.
Boost implies lifting by or as if by means of a push or other help from below, usually without the suggestion of strain or effort found in hoist and heave.