The short answer is absolutely not but it depends on “how old” we are talking about here. Every motorbike tyre comes with a manufacturing date and has a lifespan. Just like anything with a certain lifespan, it will die out at some point. Therefore, if you keep riding a motorbike with very old tyres, they are bound to give out and lead to hazards for the biker and also the bike itself.
Technically speaking, in manufacturing, every tyre is dated. This gives the lifespan in motion. Any biker can determine how old the tyres are by checking the Tyre Identification Number (TIN) on the tyre sidewall. The last four digits tell the week and year of manufacturing. For example, 1317 represents the thirteenth week of 2017 (see the picture).

Most specialists recommend stopping riding on motorcycle tyres more than five years old. However, the new tyres that are bought could potentially be several years old. So, buying from a reputed company that sells “really new” tyres.
If you use tyres older than 5 years old, you will face tyre tread, cracks, cuts, punctures, under inflation, among other hazards. So, do not cheap out on tyres and get your motorbike tyres checked frequently, and buy from reputed sellers.