Washing
Turn the shirt inside out before it goes anywhere near a machine. This protects the print, the badge and the sponsor from rubbing against the drum.
Wash cold, at 30 degrees or below, on a delicate cycle. Use a mild detergent and skip fabric softener entirely: it breaks down the fibres of older polyester over time. Never use bleach, and never wash a vintage shirt with jeans or anything with a zip.
Drying
Never tumble dry. The heat shrinks old polyester and lifts heat pressed lettering off the fabric. Instead, lay the shirt flat on a towel, reshape it, and let it dry away from direct sunlight.
Sunlight fades colour faster than anything else. A shirt dried on a sunny balcony for a few afternoons can lose noticeably more colour than one dried indoors over a year.
Ironing
Avoid it if you can. If you must, iron inside out on the lowest setting and never place the iron directly on a print, number or badge. A single pass with too much heat can ruin a sponsor print permanently.
Storing
For shirts you wear, a padded hanger is best. Wire hangers stretch the shoulders over time and leave permanent marks.
For shirts you are keeping as part of a collection, folding flat in a drawer or an acid free box is safer. Long term hanging pulls the fabric down under its own weight, especially with heavier vintage material. Keep them out of the loft and the basement: heat and damp are the two things that destroy old shirts.
If You Wear Them
There is nothing wrong with wearing a vintage shirt, and most collectors do. Just wash it less often than a modern one, air it out between wears, and accept that some wear is part of its life. These shirts were made to be worn.