No. There won't be major communist protests on the streets. Japan was very anti-communist and suppressed the communist HARD during and before WW2. They were seen as radicals that could potentially hamper military efforts and thus, suppressed and persecuted. I doubt the Japanese would suddenly say "Screw it. They killed the Emperor and now we're all commies."
It really depends on the way Hirohito dies, but the OP specified an execution.
For that to even happen, MacArthur needs to be out of the picture (and yes, I'm serious. You can't have this scenario with MacArthur). He strongly advocated for the retention of the emperor as the "head" of Japan to prevent mass staged rebellions and backlash against the American occupiers. So I'm assuming MacArthur dies in the Philippines or during the invasion of Japan. Operation Downfall would result in around 300,000-500,000 American casualties and the lives of 2,000,000-6,000,000 Japanese civilians and soldiers. That, in turn, would make the Japanese pretty much hostile to Americans due to the amount of destruction America brought upon Japan, plus the additional millions of casualties.
Now, for the actual aftermath when the scenario plays out. MacArthur is dead and Downfall succeeds after massive amounts of casualties. America and Britain are pissed beyond pissed. They plan to execute Hirohito for refusing the unconditional surrender and dragging on the war. The Japanese people's will to fight is pretty much broken at this time and with so many dead, I have no doubt that the Japanese won't be entirely against the idea. Of course, we'll definitely see pro-military and far right groups protest and rebel against the occupiers, but the execution will carry through. After the execution, I'm expecting Akihito to be Emperor, which will make the Japanese quickly forget about Hirohito and history will later treat Hirohito as a war criminal that costed Japan the lives of several million civilians and soldiers. Of course, anti-American feelings might linger a lot longer due to the amount of deaths and destruction.
(Do remember that Hirohito had to renounce his God status after the war and the Japanese people pretty much forgot about the whole "emperor is god and my soul" thing within 15 years. Though this is immediately after the war, the perception of the Japanese people will be radically different after an additional year of war and several million more dead due to the Emperor's decisions).
As for an expanded scenario. Soviets invade Hokkaido and establish a communist base there. Of course, America will strongly demand for them to return it (and they probably will, with the promise that America stays out of mainland Asia. Poor South Korea

). The Soviets eat up Manchuria, all of Korea (maybe a government in exile in Jeju, but I highly doubt this). Red China comes much quicker and the Nationalists flee to Taiwan as of OTL. Occupation of Japan and pretty much OTL, but with basically all the militarism and Shino culture COMPLETELY rooted out. In fact, after all that destruction and without MacArthur, Japan might be considerably poorer as America might be reluctant on rebuilding and assisting Japan (it'll be forced to cause the Communists, but even then). Taiwan becomes an absolute necessity against Communist Continental Asia. Japan, Taiwan, and the Philippines will definitely be seen more importantly in TL. Vietnam might be a tad messier...