Does a vow to other than Allah—if it is not an animal—become forbidden and unlawful to eat or benefit from, based on the generalization by some exegetes of the verse (And that which has been dedicated to other than Allah)?

Fulfilling a vow is only obligatory if it is an act of devotion to Allah Almighty. As for vowing to other than Allah, it is polytheism (shirk), and it is impermissible to benefit from an animal vowed to other than Allah if it has been slaughtered; rather, it should be discarded. If it has not yet been slaughtered, it is permissible to sell it or slaughter and eat it with the intention of slaughtering it for Allah. If the vowed item is not an animal, it is permissible to benefit from it in a way other than that for which it was vowed. Slaughtering for other than Allah is considered disbelief (kufr), similar to prostrating to other than Allah, as it is a form of glorification and worship due to Allah alone.