Do I need to apply for a residence permit to rent a house across Liuzhou?

No need. Official website, Housing Management Bureau of Liuzhou City, inquired that it is not necessary to apply for residence permit in different districts of the same city, even if it has been applied before, it is not necessary to re-apply across regions. Residence permit is an attempt made by some developed cities in China to learn from the "green card" system of developed countries, which has accumulated experience for China to formulate skilled migration measures and eventually form the national "green card" system of China.