Yes or no, and why?
(This is an exam question.)
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No. The sentence requires 'have' to construct the Present Perfect, which refers to a period of time extending from the past up to the present. In this case the period of time referred to is 'throughout time' which conceptually implies time without end and necessarily continues to the present.
If you remove 'have' the sentence becomes Past Simple:
'Throughout time, people always had the desire to look more beautiful and fashionable.'
The Past Simple tense is used to describe events in the past that happened during a period of time that has ended. This sentence now implies that time itself came to an end at some point in the past.