What happens when deodorants or perfumes expire
The meaning of expiry dates

The date of expiry is widely interpreted by consumers as the date after which a product is known to detiorate.  However, the meaning of expiry dates varies widely from product to product, and from country to country.  Often times, the date of expiry is not a "first known bad" date, but a "last known good" date.  In that latter case, the dates indicates that manufacturers know their product will last up until the expiration date, but after that, they don't know what happen.  Their product could last 1000s of years, completely stable and undeteriorated.  Or it could degrade instantaneously the day after expiry.  

I don't know for sure what perfume manufacturers mean by their expiry dates.  You could try asking the manufacturer for clarification.  Ask for their data from their product stability studies.

Likely routes of perfume degradation

All of the above notwithstanding, let's suppose that perfumes do degrade over a time span of several years.  Subject to that assumption, what could be happening happening?  First, I agree with the commenters that ethanol oxidation to acetaldehyde or acetic acid is unlikely to be a major contributor to perfume degradation.

However, the same is not true of other perfume components.

Oxidation


Aldehydes such as octanal, benzaldehyde, citral, are an important ingredients in many perfurmes.  Citral degradation has been studied and is accelerated by acid and oxygen, with the main ultimate product being p-cymene.  Benzaldehyde is oxidized in air to benzoic acid, and octanal is oxidized to octanoic acid.
Terpenes are also important pefume components.   Limonene is a component of many perfume blends and is a representative terpene.  It is also oxidatively unstable, with products such as limonene oxide, carveols, and carvones being formed.  Some of these oxidation products can trigger allergic reactions, which while not precisely "burning the skin", could lead to such a perception.


Other possible degradations

Perfume, especially if opened frequently, could also change by absorbing water from the atmosphere, leading to dilution and an alteration of its olfactory properties.  Frequent opening or exposure to the atmosphere could also result in differential evaporation of the more volatile components, which would also change its olfactory properties.

The bottom line

Expired perfume or deodorant might be just as good as regular perfume.  Or it might be degraded.  The degradation products might possibly include compounds that have been reported to form rashes.  It is unlikely, however, that the chemical mechanism for this is the formation of large amounts of acid.