Whitening dirty white sleeves -- along with the rest of the shirt -- doesn't always require the use of bleach. Natural products found around the kitchen, such as lemon juice, baking soda and even dishwasher detergent, are often effective at restoring dingy and yellowing shirts to their full bright white glory. Best of all, by eliminating bleach from this regimen, you reduce the chance of accidentally damaging other clothes. Try each of these methods individually until you find the one that gives you the best results.
Pour a cup of distilled white vinegar in a clean spray bottle. Pretreat dirty sleeves or any other stains on the shirt by spraying directly on the problem area. Rub the sleeves together to drive the solution into the fabric. Add another half to a full cup of distilled vinegar in with your laundry detergent when you wash the item.
Mix in roughly 1/2 cup of dishwasher detergent with your regular laundry soap to increase whitening power.
Toss a cup of baking soda in with your dingy whites. Sodium bicarbonate's mildly abrasive and absorptive properties are quite effective at scrubbing whites clean -- even removing grease and adhesives.
Add 1/2 cup of lemon juice to your whites when you launder them. Lemon juice also makes an effective pretreatment when you soak the item in lemon juice and hot water for about an hour before washing.