British citizens can normally apply for a free single entry tourist or family visit visa on arrival. British Overseas Citizens and British Protected Persons will need to get a visa before travelling to Lebanon. Persons of Palestinian origin may also require a visa before travelling, or may be required to carry additional documentation with them when travelling. Entry requirements are subject to change, so you should check with the Lebanese Embassy before you travel.
Overstaying without the proper authority is a serious matter. You may be refused permission to leave until a fine has been paid.