No. Your landlord has the responsibility to provide you with what is known as a "warranty of habitability." Feel free to Google it for some details of exactly what that means. Essentially, it means that they need to provide an apartment that is fit for human habitation. An apartment without a door does not seem to meet that standard.
Now, that does not mean that the landlord would not do such a thing.
I am licensed in Massachusetts. Any advice I provide is for informational purposes only and in no way constitutes an attorney/client relationship. If you are in need of legal counsel, you should contact a local attorney.
In addition to Adam's answer, which I agree with, in Missouri there can also be a responsibility of a landlord to provide a safe environment. Obviously, removing the door to the property would breach that responsibility and could also be a breach of the lease. However, by not paying your rent, you would have breached the lease first and would not be able to challenge the landlord on the basis of breach. That being said, the landlord must go through the eviction process, including providing you notice of the eviction, and can't just remove the door because you are behind on the rent. If the landlord has not yet provided you with notice of eviction and begun the eviction process, she would likely be liable for any damage or loss to your property if she does remove the door. Much of the determination of breach is governed by the lease, but the bottom line is that she can't just remove your door.
Anytime a landlord ousts a tenant WITHOUT going through the judicial process, and I mean ANYTIME (even when a tenant hasn't paid rent). that landlord is liable for forcible entry and unlawful detainer. A landlord cannot reclaim possession by changing locks, removing any personal property, or by removing the door. Even if you did not pay your rent you still have a claim for forcible entry and unlawful detainer under RSMo. section 441.223 and RSMo. chapter 534.