It depends upon whether or not you are fully vested in the shares. Under IRC sec 83(a) you are taxed on the fair value of the shares minus what you paid for them (if anything) in the first taxable year in which the shares "are transferable or are not subject to a substantial risk of forfeiture, whichever is applicable."
See IRC 83(a) at https://www.law.cornell.edu/uscode/text/26/83
If you aren't immediately vested, then if the startup shares significantly appreciate when you do vest you would suddenly have a big tax bill even if you cannot sell the shares. (e.g. no IPO, lockup, etc.). People will often make an 83(b) election to be taxed when they receive the shares. Why? Because the value is really low then and the tax is nominal. After that they will only be taxed on the gain if and when they actually sell.
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