Kevin Ryan suggested "I don't support the war, but I do support the troops" is an example of false equivalence. I eventually selected that as the right answer, though I wasn't 100% certain.
Right or not, I don't think it's the whole story; I think this particular propaganda package is a little more complex.
First, I think it includes what you might call an implied or subliminal appeal to authority. Propagandists have expended enormous energy in promoting the idea that military personnel who commit heinous acts aren't guilty if they're only following orders. Of course, those orders come from above, with the commander-in-chief at the top of the food chain.
If the troops are absolved from guilty by merely following orders, then their superiors are also absolved from guilty, because they, in turn, are following orders. (I think this might be an example of what philosophers call regression.) In the end, the only guilty person is the commander-in-chief.
Yet even he may be ultimately protected by this line of reasoning. If the president is seen as doing the bidding of corporate entities, bankers or whatever, then he may also be seen as simply following orders. The guilty party is now essentially invisible.
The propagandists also tap into appeal to common belief (aka argumentum ad populum). By giving lots of publicity to citizens (particularly liberals) who obediently say "I support the troops," they send out the message that everyone supports the troops.
Another consideration is the fact that even many, if not most, anti-war protesters have relatives who are in the military. Protesting a war when your cousin is in harm's way can be extremely uncomfortable. When fellow (would be) protesters chant "I support the troops," that makes protesting a war even more painful. One could also argue that protesting might inflict psychological pain on the troops, some of whom might really believe they're doing the right thing. Catch-22.
Yet another propaganda technique is the mantra that the military is essentially necessary or good. In fact, one of the answers to my question begins, "Military personnel are trained professionals who put their lives on the line for the public good."
That isn't true. There are many grossly unprofessional people on the military, and many military personnel never risk their lives. One could argue that they haven't done anything for the public good since the War of 1812. Even people who think they can cite exceptions would have a hard time making the case that the war in Vietnam or the destruction of Libya helped the averge U.S. citizen.
Finally, there's at least one more element in play, but I'm not sure what to call it - maybe obfuscation or confusion?
If a group of anti-war protesters are interrupted by a person (particularly one who claims to be a liberal) who says "I support the troops," they will likely be cowed into silence. Exactly what does that mean, and why would a fellow liberal interrupt our protest with those words? If we climb on board and start cheering for the troops, how does that not support the war effort?
In summary, the words "I support the troops" don't technically qualify as a fallacy. But if you change it to "I don't support the war, but I do support the troops," then we're talking about a borderline fallacy at the least.
When you look at the whole package, you see a variety of strategies based on philosophy and psychology that serve to confuse and intimidate anti-war protesters.
Kevin Ryan's answer, false equivalence, helped me put the pieces together. It may also be a better answer than any of the individual points I discuss in my answer, so I'll leave it at the correct answer. However, I don't think it's a complete answer.