What would happen after transforming competent E. coli cells with a plasmid without an ori?

The E. Coli will replicate, but the plasmid containing your antibiotic resistance gene would remain in one of the two daughter cells without being replicated. So basically, your population of transformed cells will remain constant while your wild-type cells continue to grow in number.

Bearing this in mind, it's impossible to "grow" resistant colonies under these conditions because the size never changes. Selection experiments are therefore difficult to impossible, unless you succeed in finding your resistant bacteria under a microscope!