A lot depends on the cell line, efficiency of transfection, seeding density, and what else you are expressing in your plasmid. For transient transfections in rapidly dividing cells (less than 24 hours), you have a window from about 2 to 7 days...if the cells divide more slowly, then the time can extend. Reductional division of your plasmid will depend a lot on what is written above. If your cells express LaT, then most likely longer as they will be able to replicate.
Differently to whats happen in bacteria when after transformation, thanks to the presence of replication origin region and antibiotic selection markers the plasmids are maintained and propagated (passed from cell to cell) also during the bacterial replication, In transient transfection, the introduced nucleic acid exists in the cell only for a limited period. It is not passed from generation to generation during cell division, and it can be lost by environmental factors or diluted out during cell division, therefore after transfection the cells are maintained in culture for several days (the length depends from the cell line, growth media and conditions) but no dilution and passages were performed, and a new trasfection have to be performed in empty cells, that are propagated in parallel to the trasfected one, when you would like to produce more protein.
If you would like to have an idea of a standard protocol, you cah see the following video
I correctly infer from your question, it refers to transient (unstable) transfection which does not involve any gene integration. Firstly, not all cells in your culture media get transfected (since the efficiency can never be 100%). Therefore, one deals with a mixed set of cells, ones that receive the plasmid and others that do not. It will thus not be ideal to subculture this population for further experimentation until you make use of a selection marker that eliminates the un-transfected cells. Secondly, in this case, since the transfected plasmid (pcDNA 3.1) remains as an extrachromosomal entity (episomal), the plasmid will never be stable and will ultimately be lost during the course of cell division and growth.
In summary, such cells should be grown/maintained for only 5-7 days post-transfection. But, it is not recommended to passage transiently transfected cells. In case you wish to do so, make sure you supplement your medium with a suitable selection marker (antibiotic). My advice would be to transfect a fresh batch of cells each time for performing any experiment.