Consider:

Corp is using Tennin Institute, and has five servers: The three centrals, and two remotes. One remote has a bunch of stuff, and the second has an Asset protected by a single piece of Ice.

The runner runs that second server, and the Corp does not rez the Ice. The runner finds the Asset inside, and trashes it. Now, the server is a piece of ice over nothing.

Runner plays Security Testing.

Can the runner designate that second remote server as the Security Testing target? Can a run against that fort truly be considered "successful" since there's nothing there? Does the Corp in this case have any recourse besides rezzing that ice and putting more ice on this empty server? Does Tennin Institute ever have another workday again?

Unless the ICE on that remote is destroyed, the server exists and runs against it will be considered successful (despite the fact that no cards are available for access) if they manage to get that far.

With Tennin you have to be more careful about making it too easy for the runner to get successful runs and empty or unprotected servers are one such thing that is perhaps best avoided (particularly if Security Testing is a possibility as it can negate some of your traps by avoiding actually accessing them).

The one corner case to note is that [an unprotected] Jackson Howard, if removed from the game [during a run], the server ceases to exist and the run is considered neither successful or not (see the FAQ).

Edited for Clarity.

It seems unlikely, but weird stuff is why this subforum exists, so. In the above situation, does the Corp have the ability to trash that isolated Ice when he installs a piece of Ice elsewhere, or does that only work if you install on the same server as you remove from?

Only on the same server.

But the Jackson Howard thing is common enough that the conventional wisdom is that one ought not do a Dirty Laundry run on a single unrezzed card in a remote.

Indeed, weird stuff is what makes the game interesting.

Corp ICE trashing is only really available to reduce the cost of installing onto a server, so it doesn't quite fit. The closest you can get on the trashing front is to use a lot of unique ICE that will trash previously rezzed copies when two become active (although it would be a bit tricky to have this trigger during a run and cause the remote to "Jackson Howard").

If deleting remote servers with just ICE becomes really necessary I guess you could use that HB executive that allows you to bounce installed cards back to HQ for a click (so again, not for mid-run).

In my opinion, Tennin prefers a playstyle more in line with "Big ICE" (servers with deep and/or costly ICE) and un-protected remotes for as you correctly identified ICE only remotes will make it a bit easy for the runner to deny Tennin's ID ability (but perhaps is making them otherwise waste clicks - and sucessfully running isn't always in the runner's best interests for self-preservation).

Another cornercase example is a Server with no cards installed in it and a single Data Mine protecting it. If the Runner doesn't break the subroutine on the Datamine, the Server will cease to exist mid-run.

If deleting remote servers with just ICE becomes really necessary I guess you could use that HB executive that allows you to bounce installed cards back to HQ for a click (so again, not for mid-run).

Also Security Subcontract (Core Set) - again, costs a click, so no mid-run.

The only card I can think of that can do this mid-run is a rezzed Himitsu-Bako.