The timing and movement of prisoners to the concentration and extermination camps did significantly impact Germany's ability to supply the eastern front. Allied intelligence noted this on many occasions and it has been speculated that Allied reluctance to interfere with "transports" was partly due to the negative impact they had on the Nazi war effort.
The big movements of people were in 1942, during the year or so the extermination camps operated, then there were the other times when the trains were given priority over other traffic, which the Heer complained about, but not a significant amount of lost cargo capacity overall. In October 1941 the Germans planned to have 724 trains arrive at the Dnieper supply base, and received 195. The allies had trouble interdicting the French and Italian rail systems, they did not have much ability to do so in Poland or Hungary.
At least one refinery at Ploesti could produce 90 octane aviation fuel according to the various reports. The USSBS says direct exports Romania to the German army in the east in tonnes, 1941 34,351, 1942 369,452, 1943 715,749. In 1943 Romania represented about 62% of crude oil available to the European Axis. However it had built an economy using that oil, in 1938 6.6 million tonnes of crude, 1.7 million tonnes of oil products used domestically, 1943 it was 5.27 million tons of crude, 2 million tons domestic oil consumption. Oil shipped to Germany normally used barges on the Danube, the allies mined the river.
Rail was the chief mechanism to supply the German army in the east, though ships were used on the Black Sea and Baltic. The Reichsbahn ran the rail system, so maybe look up its histories. No information on the size of bulk fuel shipments and when/where they were broken down into smaller shipments/containers. There does not seem to have been many German tanker trucks. Truck supply in the USSR had the well known problems of lack of trucks and bad roads.
The initial 1941 advance gained from captured supplies, but the main lines of advance were not always along the rail lines, leaving them to be cleared by follow up troops, which slowed down bringing the line into service. The Soviets were quite successful in evacuating rolling stock, sabotaging signal systems and track. Apart from the gauge issue one other Soviet difference was their locomotives were designed to go for longer distances before needing water, so fewer water points, a problem for the standard gauge locomotives.
As the war went on partisan operations against the rail lines the Germans had in service increased.
It is quite old now and probably should be considered as an introduction but check out Supplying War by Van Crevald.
The multi volume set Germany and the Second World War will have some more details.
For an idea on the overall fuel situation, Oil and War by Goralski and Freeburg. In 1940 Egypt had an oil output of 17,773 barrels per day, that is more than Germany. Its capture, assuming the same 1943 production, would have added another 14% to axis crude oil supplies.