Words like move, remove, prove used to be pronounced with the long [o:] sound before the Great Vowel Shift (which is how they got their spelling), and by Shakespeare's time they moved to [u:], which they retained until modern times. I tried to find any evidence other than the love- rhymes that these words could be pronounced with a short [ʌ] sound, and couldn't.
On the other hand, "love" was probably pronounced usually with a short [u] sound, the same sound we use today in "put". All the words with u in a closed syllable used that sound as well, like "cut", "but", "cup" "sun" etc. The change from this [u] to [ʌ] happened after Shakespeare's time, mostly later in the 17th century, resulting in the familiar pronunciation of "but" "cup" "sun", but leaving after it a fair number of exceptions that refused to follow the change, such as "put" "push" etc.
"prove" and "love" did rhyme when Shakespeare used them; but perhaps the rhyme was a bit inexact, rhyming long [u:] of "prove/move/remove" and short [u] of "love" (but still much more exact than modern [u:] and [ʌ] of course). Or, as Charles Barber claims in his "Early Modern English" (unwieldly Google Books links: http://bit.ly/1uI7SWQ), "love" was exceptional in having a variant [u:] pronunciation especially beloved by poets, precisely because they could use it to rhyme it with handy words like "prove" and "move". Either way, the love/move rhyme did not sound at all the way Ben Crystal pronounces it.