A tire won't cause vibrations on startup. I'd suggest loosening or removing the fan belt on a cold start to see if the shake is internal or external. A slightly bent fan blade can cause some oddball harmonics at various engine and/or vehicle speeds, and some types of tires can develop flat spots when they sit for extended periods of time- - - -the dreaded "nylon thump" that early synthetic cord tires used to have. I've seen those things have teeth-rattling shakes for 5 miles or more on a cold day!
Jerry

"It is better to be silent and be thought a fool than to speak and eliminate all doubt!" - Abraham Lincoln
Cringe and wail in fear, Eloi- - - - -we Morlocks are on the hunt!
There is nothing noble in being superior to your fellow man; true nobility is being superior to your former self. - Ernest Hemingway
Love your enemies and drive 'em nuts!