I almost never see violins with a fine-tuner on any but the E string.
Is there any drawback to placing and using fine tuners on lower strings? (Other than the need to use strings with loop ends.)
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There are a couple of drawbacks to using add-on attached fine tuners.
The violin tailpiece should be set up so there is a specific distance from the bridge to the contact point of the tailpiece to facilitate a resonant response in the strings. The average recommended distance is 57mm, although some Luthiers will set the distance by tuning the strings with tail-gut adjustments for best tone. Using the extended arm fine tuners will make the distance from string end to bridge too short, affecting the instrument's tone.
Adding mass to the tailpiece may also have an affect on tone as well. I haven't personally tested this.
Using the screw down clip-on style fine tuners (the tuner pushes a screw against the string to deflect it and make it tighter) can cause early wear-out of your strings, as well as shorting the distance to the bridge.
For students, the ease of tuning (especially considering the peg setup of student quality instruments) usually outweighs the loss of tone.
Integrated fine tuner tailpieces alleviate the problem of shortening the string tail length, and most are made to accept ball end strings, so as said in the comments there isn't any reason to not use one.
Fine tuners don't work as well with low tension or gut strings because of the amount of motion needed to change the pitch of the stretchier strings, but they work fine with modern Nylon based and steel type strings.
It's a bit pointless with non-steel strings since the precision of a smooth-moving peg is quite adequate and their range is required since non-steel-core strings hold pitch worse and react to length changes quite less. When moving a fine-tuner too far, you'll damage the violin's top (depending on tail piece and fine tuner construction), so reserving fine-tuners for just the string(s) where they are by far most useful is a good compromise. Also they used to be a noticeable expense.
That being said, I've seen enough violins/tailpieces that apparently came with (identical) fine tuners from the start.