While trying to figure out how to rearrange products on my catalog/products page, I found this article and when I created a new collection to overwrite the default catalog/product page, I noticed it cleared out unused tags.
http://docs.shopify.com/support/your-store/collections/taking-control-of-your-catalog-page
Directions.
First remove all the tag(s) you don't want from every product.
Next create a new collection with the url handle of all. (This will overwrite your current catalog/product page. See directions in link above under step 2.)
Now the unwanted tag(s) you didn't want will no longer be in the browse drop down menu.
If you have more than one collection, I imagine you would have to delete/recreate each collection as well. I'm not sure since I only have used the main catalog/product page.
This did not remove the unwanted tag from the list of recently used tags though. It only removed the tag from the browse drop down menu.