HELP! My vintage tulip side table is covered in paint!  

Hmmmm-pretty delicate operation
The tulip base has a coating of sorts,which a paint thinner would dissolve.You might consider stripping and re coating it entirely.
As for the wood surface,you might try fine 0000 steel wool and paint thinner,followed by linseed oil,to nourish the wood.Have plenty of terry cloth on hand to absorb and remove the paint,too.
Good luck!

how would i go about...
how would i go about recoating the base once it's entirely stripped? what type of wood should i expect to see under the paint on the top. walnut or rosewood? should i apply a satin poly to the top, or go sans protective coating?

You might try...
giving it to a friend...;0)

i'd love to, and considered it, but....
it's restoration will allow me to keep the four matching armshells!!!!!!!!

Powder coating
Most cities have shops that do this kind of thing routinely. Check Yellow Pages or equivalent. Wide range of colors typically available. Just like the guys in the link (No endorsement intended or implied).
http://www.custompowdercoating.com/

alcohol
Before you do anything else, try a little bit of alcohol on the underside of the base at the top. If it's latex paint, alcohol will remove it without harming the original finish. I have a vintage Saarinen table whose base had been repainted long ago and the paint came off pretty easily with just alcohol. The original finish is matte and ecru in color, which i've heard is original but not sure if that's accurate.

ecru
For what it's worth, my little Saarinen side table base (made in 1969) is finished in what I'd describe as a sort of warm white or cream shade. No PMS chips handy for a scientific match, and no way to sort out the longterm effects of sunlight, tobacco smoke, and oxidation.

No worries Geo...
you'd be hard pressed to find that color in a Pantone deck anyway, you'll have to create your own CMYK mix.
;0)