No, you can't leave it out. If the sentence had 'needs gain' it would make 'All tyranny needs gain a foothold' the main clause [="All [kinds of] tyranny needs [to] gaina foothold'] ; and the 'sentence' would not make sense.
What it does mean is 'For tyranny to gain a foothold all that is necessary is that well-meaning people remain silent'. Jefferson's version is neater.
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