Birch is light, inexpensive, easy to work with and is used for the wood veneers on most inexpensive furniture. However, birch plywood is difficult to finish. The different layers of wood may take stain differently, resulting in uneven colour. It's not impossible to stain birch plywood; it just takes extra prep work to get an even finish.
Test the following steps on scrap plywood to make sure that you get the colour you want before working on the actual project.
Give the birch plywood a light sanding with 220-grit sandpaper. Work with the grain, not across it or in circles. Wipe off the dust with clean shop cloths.
Apply a pre-stain conditioner formulated for birch. This is available at home improvement and paint stores. It is usually a spray, but can also be applied by hand. Let it dry according to the manufacturer's instructions. This pre-treatment will help the birch plywood take the stain more evenly.
Dip your staining sponge into the tinted wood dye and apply it to the birch plywood. Work with the grain, as you did when you were sanding. Make sure to feather the edges as you re-dip the sponge to avoid any thick "borders" between areas as you stain. Let the stain dry according to the manufacturer's instructions.
Apply a second coat if needed, and let it dry as specified.
Put on your painter's mask, and apply a coat of sealer. Polyurethane spray is the easiest to apply and the longest-lasting. Make sure that whichever sealer you use is compatible with your wood dye. It's best to stay within the same brand. Let the sealer dry as the label suggests, and then apply a second coat, if you like.
Use a small piece of masking tape to label the back of your scrap wood if you are experimenting with different colours.
Always work in a well-ventilated area when using spray sealer.