It looks like the short answer is "it depends"... ie. nobody can know if the articular testing protocol followed by your workplace will have a chance of a job-threatening false positive from the hand sanitizer supplied. You need to know how the test is used.
The people to talk to would be the people doing/administering the test - you are concerned that routine hand sanitizer use could lead to a false positive in your required drug testing - can you get a clarification on this. You can also check with a union rep to see if there is a problem with this sort of thing.
There is a possible downside in which the possibility that some people, worried about a legit positive, may want to muddy the issue by using hand sanitizer - introducing a plausible deniability - and leading to people with a genuine concern about false positives getting tarred with the same brush.
Sometimes these tests are available commercially as a small kit - in which case you can do some experiments specific to your workplace, to see if your concerns can be supported by direct evidence.
You still need to know how the test is administered though. For instance, if the workplace has a zero tolerance policy, then the slightest trace may be bad, but if they just want to make sure that employees in your position are not alchoholics or routine drug users, then small traces are unlikely to be an issue ... a positive test may just result in an investigation, it's a matter of policy.
If this is a serious concern, especially if you are not the only one, then maybe there is a case for switching to an alchohol-free hand sanitizer?
How does this come up? Is this a new policy? How long has the hand sanitizer in question been used in the workplace while there has been this sort of testing going on? How long have you worked there? Etc.
But have you noticed how the responses are much better now that your actual concern has been voiced?