Medium answer : Lactate is actually a buffer that helps to alleviate the burn, in fact. However, as it is always associated with the burn, it becomes an innocent bystander that is guilty solely by association.
Long Answer :
Glucose within cells is broken down via a sequence of 10 chemical reactions (glycolysis) that ultimately produces two pyruvate molecules. The reasons that a cell would want to do this are twofold : first, the production of each pyruvate molecule yields one ATP molecule ( the energy currency of the cell ), and second, this process is extremely rapid, so even though you don't get much net energy out of it, your energy production engine, though short on horsepower, has a really high rev limit.
Pyruvate is in itself a valuable end product because it can be further processed in the mitochondria via the TCA cycle, where each pyruvate generates an additional ATP molecule. Other compounds are produced as well via the TCA cycle, including 4 NADH molecules and one FADH2. These last two compounds are very important because they feed the electron transport chain in mitochondria (ETC).
For each NADH molecule, three protons are generated via the ETC. The final electron acceptor in this chain is oxygen. For each proton, the mitochondrion membrane proteins produces one ATP molecule. So, for every NADH molecule that we manage to pump through the ETC, we're going to get 3 ATP molecules. Since each pyruvate yields 4 NADH, we get 12 ATP per pyruvate, and since each glucose molecule generates 2 pyruvate molecules, we walk away from the ETC with 24 ATP molecules! Not bad for a day's work.
Each FADH2 gets us 2 protons, and therefore 2 ATP molecules per pyruvate molecule. Net, net then, each glucose gets us 4 additional ATP molecules from FADH2 processing via the ETC.
Totting up, if you've been keeping track, per glucose molecule we get 2 ATP from glycolysis, 2 ATP from the TCA cycle, and 28 ATP from the ETC. That's 30 ATP from the mitochondria, and 2 ATP from glycolysis, which kicked off the entire process in the first place.
While there is an awful lot of energy derived from the ETC, there are drawbacks ... one, you must have available oxygen to accept the final electrons, and two, this process is much slower than glycolysis.
So what happens in cases where there is insufficient oxygen to handle the electrons being generated via the ETC?
Well, things start to look a bit like this (at about 4:40):
The ATP production line gets overwhelmed, and instead of producing ATP and water, as is normally the case, it short circuits in an attempt to get faster, and produces much less ATP, and instead of water, you get reactive oxygen species like superoxide or hydrogen peroxide. The good thing about these is that they accept the electrons and quickly react with whatever is around them, thereby releasing the oxygen again to go back to the ETC. The drawback, of course, is also precisely that they react with whatever is around them, causing damage in the process and giving you that tell tale burn.
Meanwhile, as the production line gets overwhelmed, pyruvate from glycolysis starts to build up. As the concentration of intra-cellular pyruvate start to increase due to its inability to enter the mitochondria for further processing, an alternative pathway starts to get invoked which converts the pyruvate into lactate. This helps to buffer the reactive oxygen species that are generated by the mitochondria, as well as being shunted to the liver for the regeneration of glucose.
The really interesting thing about this is that once you start feeling the burn, your muscle cells are grossly inefficient in terms of glucose utilization, yielding about 2 ATP molecules per glucose molecule instead of the usual 32 ATP molecules. Or in other words, you now require roughly 10 times the glucose to maintain your activity level / energy output ... you are crushing glucose levels.
If you do this repeatedly, your metabolism has no choice but to adapt or die, so you will adapt, and one of the adaptations will be that your cells will upregulate various oxygen transport proteins, making those cells that adapt in this way much better at sucking all the available oxygen from your blood stream!