Consequences of applying to both research and staff positions at the same school simultaneously
At the same school, it would depend on how closely related those two positions are and how likely the same person is to see the same positions.

Let's say you are a biomedical scientist. Let's say that Professor Smith is hiring a postdoctoral researcher in your field, but the Center for Awesome Research that he works in is also hiring an administrative research role (let's say a grants coordinator) that you're interested in. The chances that Professor Smith sees both of those applications is significant, although how high it is depends on how involved in the administration of the Center he is. This could be bad. Professor Smith might think that you aren't really serious about research if you are applying for administrative roles. Likewise, the person hiring for that admin role might think you're only applying as a backup and that you really want to be a scientist. Net result is that you get neither interview.

(Now, of course, that might not happen; Professor Smith might not really care because you are excellent or reason that you are simply trying to keep options open; the admin position hiring manager may decide that this job is your true passion. Or neither may see the other's application. But it's a risk.)

Now, let's say that you apply for a biomedical research scientist position at the medical school but you also apply for a position in institutional research at the main campus. The chances that the hiring managers for those two positions will find out that you applied for the other one is probably really close to zero. So that would probably be ok.

In general, you should probably try to apply to those admin/staff positions at different institutions from where you apply for the scientist roles, but applying to admin/staff positions that are pretty separate from the science roles in terms of administrative structure would probably be fine.