That would explain the issue! A few options:
Yes that is certainly one option. The disadvantage is that including all eukaryote sequences would make a much larger and slower classifier, when you presumably don't care what the other euks are, you just want to get rid of them. If you know what non-fungal reads you expect to find, you could just selectively add those to your database to provide an "outgroup" for identifying those non-fungal reads.
Option 2 is to just take the data you have now and filter out all unclassified/phylum only sequences with qiime taxa filter-table. The advantage is that you don't need to re-classify again, the disadvantage is that you are assuming that these indeed represent non-fungal seqs.
Option 3 is to use qiime quality-control exclude-seqs to filter out sequences that do not align against the UNITE fungal ITS database. The advantage is that you are explicitly filtering out anything that does not hit within a specified % identity, the disadvantage is that it adds more time.
I'd personally try option 2, see what I'm left with, and then maybe move onto option 3 if I'm still harboring some doubts.