Where does the lactate produced by tumors go?
Most of the lactate produced by tumors probably enters the circulation and is taken up by other tissues (like the Cori cycle, correct). There are several studies like this one measuring glucose uptake and lactate release by tumors in vivo, and it is clear that a substantial amount of lactate is released into the veins draining the tumor site.

Oxidation of lactate by neighboring cells within tumors has been reported as well, but I don't know of any quantitative data on exactly how much lactate this process can consume. But it seems unlikely that this can remove a major fraction of lactate produced by glycolytic tumors, since glycolysis produces lactate at a much higher rate than respiration consumes it (per ATP generated).

It's also true that tumors can accumulate lactate. Some tumors are poorly perfused, with "disorganized" blood vessels and blood flow, and then waste products may not be cleared away efficiently.

So yes, all of the above :)  There are large differences between individual tumors in all these parameters, so there is no generally valid answer I think.