Does ESR rise in MS?

No. Sed rate (erythrocyte sedimentation rate / ESR) is a famously poor guide to the presence or absence of disease. There are a few rheumatologic diseases in which it's almost always high and this helps rule out ordinary aches and pains. And the test is easy to do wrong. Don't set much by this lab result, despite the fact that there's inflammation in MS plaques in the active phase.