A study came out March 13, 2008 in Neuron, linked above (Gibson et al. 2008) in which the researchers found a link between capsaicin and long term depression (LTD) in slices of rat brain. They found that the capsaicin receptor, TRPV1 plays an important role in excitatory synapses of the hippocampus. LTD is one of the processes (along with its opposite, long term potentiation, LTP) which the hippocampus uses to make new memories. Simply put, the ratio between LTP and LTD in the hippocampus is what decides whether you will remember something or not. LTD is a process of weakening connections between neurons, making that neural circuit less likely to activate, while LTP is the opposite, strenthening connections. The authors claim that TRPV1 agonists (capsaicin) induce LTD, while antagonists (capsazepine) inhibit it.
This may have implications for some drugs containing capsaicin (such as the anti-obesity drug Acomplia). It may also open up research into new drugs, one of the diseases that could benefit from this research is epilepsy.
Peripheral application of capsaicin is unlikely to have any memory related side-effects, nor is ingestion, because it will never reach the central nervous system (CNS). Application to the skin is just that, to the skin. It will act on peripheral pain neurons, but will never come in contact with the CNS. When capsaicin is ingested, it quickly gets broken down by the intestines, and if some does reach the blood it will be metabolized by the liver. There is no reasearch which indicates eating lots of hot peppers is bad for memory, so don't worry.