Is it permissible for someone who lives in a house they do not own, and for which they do not pay rent, and whose owner repeatedly demands it from them, to go for Umrah, while they are slowly building another house, despite being able to complete it quickly?

We do not know the nature of this person's dwelling. However, if he is residing there by usurpation, it is unlawful, and he must repent, vacate the house, and guarantee its usufruct. If he is residing there with prior permission, he is not permitted to continue after its owners demand its evacuation. He must fulfill his religious obligations, such as Hajj and Umrah (definition).