There would be a caretaker council officially in charge of the government, perhaps with Molotov as a first among equals. In actuality, the Red Army will likely get more and more power, but whether they do anything with it during the war is unknown.
In terms of impact on the war, Stalin more or less had a negative impact in 1941-1942. He made all the mistakes, and continued to make them up to Case Blue. His major "credit" for victory was finally accepting he was wrong and stopped overruling Zhukhov and STAVKA. So the question here is whether the troika or whatever that now rules in Stalin's place will make worse decisions than Stalin, or if they simply let the Red Army do their own thing. I believe it's likely to be the latter.
This means 1942 is initially a lot bloodier for the Nazis, but they avoid the disaster at Stalingrad because they aren't overextended. Ironic, isn't it? I'll say that it's a wash by end of 1942 except that the Germans and their satellites have more manpower. It's obvious they can't win, but they still hope that a victory in 1943 will result in a negotiated peace. That fails just like IOTL. The war continues more or less as we know it.
However, there are now huge changes in the postwar period. Stalin was really the person who insisted on Soviet policies that caused the Cold War. He chided even Molotov for giving into the West too much in 1945-1946. Without Stalin, there is no force that keeps pushing everyone into confrontation. There will be disputes of course, but a working relationship is at least possible. There is likely to be a de-Stalinization period soon after WWII, a grander demobilization of the Red Army, and more resources put into the economy.
- Czechoslovakia may keep its democracy.
- NATO may not be formed.
- Soviets may enter some kind of economic arrangement with Bretton Woods, GATT, or allow Eastern Europe to enter the Marshall Plan.
- No Stalin-Tito split, and Yugoslavia becomes an ally of the Soviet Union.
- Some form of US economic aide may be given to the Soviets, especially in return for cooperation on letting Eastern Europe be free.
- Poland will remain a major thorn between the Soviets and the West, but other countries might be let go provided they are sufficiently neutralized and pose no threat (like Austria or Finland) which means Hungary, Romania, and Bulgaria become relatively free by the late 1940s.
At some point, there may be a reactionary push back to Stalin's hostile policies, but I think the greater prestige of the Soviet Union, good relations with Britain and the USA, a focus on the defeated Germany, and beneficial economic relations will win out. Still some disillusionment over what everyone hoped postwar cooperation would be, but workable.
There will likely be some sort of falling out from inevitable decolonization and the situation in China, but there will be no Cold War nor any hot wars like Korea. It'll be a very different postwar period.