Is it permissible for a father to swear an oath upon his son not to enter the house if he travels to seek better work opportunities, and what should the son do in this situation?

If there is no harm to you or loss of benefit in obeying your father by establishing the project, then you must obey him. However, if there is harm to you, or if your benefit lies elsewhere, then there is no blame in not obeying him. A parent does not have the right to prevent his mature child from managing his own money or striving for lawful earnings. You must apologize to your father, treat him with kindness, and seek his رضا (رضا: satisfaction/pleasure), for the father's right is great, and the Lord's رضا (رضا: satisfaction/pleasure) is in his satisfaction. And the father is the middle gate of Paradise.