Oh, this is a great question!!!!!
Yes, generally it's possible - and in fact quite easy - to change a white fabric to a light colour eg light tan. If you have a photo with your white blanket in it, please post it so I can show you how to make the change.
A few caveats:
Make sure the white blanket isn't overexposed. There needs to be visible detail in it. Otherwise yes, the change will look fake.
You must only expect to change from white to a very light colour. Pale pastel type colours, you know? If you attempt to move further, to dark or vivid colours, the fakeness sets in.
Remember that this change will involve masking. So be very very careful not to make the masking too complex. Make sure it's a blanket with straight edges, not frilly edges; and make sure anything on top of the blanket (eg other blankets, or clothing) are also not frilly.