Do conductors age under reasonable electric load?
If the question is about small transformers, the above answers are adequate.

If the question is about large transformers, i.e. oil-immersed power transformers - that is a whole different game.

The metallic conductors in a transformer don't deteriorate with age, but the insulation does. Power transformers are constructed with paper insulation, which has a limited lifetime. The cellulose in the paper degrades with normal heating and becomes mechanically weak; additionally, the degradation of cellulose releases water, which lowers the di-electric strength of the oil.

The rate of degradation is exponential with increasing temperature (see 1 for some discussion.) I believe transformers are usually designed to achieve 25 or 40 year lifetime under rated load.

All of which is to say - the metallic conductors in an old power transformer will be fine - but beware old insulation.