How vibration sensitive are hard disk drives?
Laptop drives are generally more vibration-resistant than desktop drives (they are designed to be used in a moving plane, car, in a laptop on someone's laps, etc.), so from this standpoint there shouldn't be a problem taping them somewhere inside your computer's case. There are other problems, however:
When hard drive is fastened properly, metal bracket also acts as a heatsink. Moreover, in a properly designed case there should be an airflow around a hard drive. Your tape will have lower thermal conductivity than metal, and depending where you tape it, there could be less airflow available. So your hard drive could overheat.
Constant vibration (from hard drive's motor, computer's fans, etc.) can loosen the tape over time, depending what kind of a tape you use, it can dry up, etc. So don't put your hard drive vertically somewhere, make sure it's fairly secure without the tape, and use the tape as an additional measure, not as something that prevents your hard drive from falling down on your motherboard, for example.
Make sure you don't short anything on hard drive's PCB, if it is exposed (e.g. there are no protruding screw heads on the surface that you tape your hard drive to).
I've used tape to secure 2.5" hard drives in a desktop tapes, but only temporarily (till I got around to order a few of those conversion brackets).
The hard drive is not particularly sensitive to vibration -- vibration will destroy any piece of electronics over time, but drives produce their own vibration and hence are "beefed up" a bit. And laptop drives even more.
The real limit with hard drives is peak G force. This isn't vibration so much as "shock" -- a sudden jolt -- and it only takes one such event in excess of the drive's G limit to damage a drive.
A bit of a war story:
Some years back, when I was working on IBM AS/400 computers, they were having trouble in the development lab with drives failing. To permit multiple people to (serially) use a single test system, drives were mounted in sliding plug-in trays, and drives would be exchanged when a test scenario was changed.
But it seems that drives would fail when exchanged this way -- a drive would be good, would be carefully removed and gently set down on a conductive foam pad. But then, an hour later, the drive would be inserted and would be found to be "crashed".
Analysis of the drives showed that they'd been subjected to excessive shock, but originally no one could figure out the source of the shock. Then it was realized that the shock of snapping the drive into the socket (as the tray was slid into the machine) was enough to do this damage.
Of course, these were old "big iron" drives, and current desktop and (especially) laptop drives are much more robust, but still the hazard is mechanical shock that does not seem, to a human, to even be worth noticing, but which to a drive can be devastating, due to the proximity of the shock source to the fragile parts in the drive.