How much? A lot, or what you can only see with a tuner? Pressing firmly with your thumb doesn't (normally) bend the neck, but it can stretch the string (especially the higher strings, and especially if you use light gauge strings) over the fret and pull it sharp. A guy in my band who has played 12 string acoustic for many years has to use pretty heavy strings on his Tele because he is used to fretting very firmly and pulls notes way sharp on a six string electric with light strings. Tall frets and high action make this more pronounced, too.
Also, most guitars flex some; I can pull my LP a little sharp by pulling the neck back against my midsection. That's normal.