What is a lower bound on the vocabulary size for generating word/sentence embedding using word2vec or skip thought vectors?
It's not a straightforward question to answer as it is hard to compare the quality of two word2vec models with a meaningful metric. You could, of course, use the loss function, but that won't give much. 

Another approach is  more heuristic: take for example the frequency of each word, and remove those words that are repeated less than N times, where you can set N to for example 10 or 20. This is common practice as you need a certain number of repetitions of the same word to have some meaningful results.