You mean Planck length?
A physical door is made up of atoms that vibrate and stuff, moving many planck lengths around all the time due to their temperature whether you open it slightly or not. Moving the atoms near the end by a planck length would make an insignificant contribution to it.
To me it sounds a bit like you're describing a mathematical door? A perfectly thin, continous line, in that case, the line between the endpoints would move within the planck length, but remember the particles in the physical door would fluctuate all around the line and not be locked to it.
I haven't started on quantum mechanics yet (starting on it in November) so I don't know all that much about the properties of the planck length. But I don't think it's necessary to answer the question, depending on what you precisely mean ofc.
More fundamentally (or what you probably intended to find out), your question sounds like 'can anything move a distance shorter than the Planck length'?