Functionality of aptamers depends upon the oligo's sequence and secondary structure. So, if I take a DNA aptamer and make an RNA oligo of the same sequence (T replaced with U obviously), will this RNA oligo also act as an aptamer?
There is at least one known case where both the DNA and RNA versions of the same aptamer sequence bind the same target (Lauhon and Szostak, 1995).
But you can't generalize this, there are two differences in structure between RNA and DNA, the Uracil/Thymine change and the missing 2'-OH in DNA. Those differences can affect the binding and overall structure of the aptamer.
It is possible that the RNA version of a DNA aptamer binds the same ligand, but this won't be always the case.
Aptamers are single-stranded sequences that bind to proteins or other molecular partners. Obviously, ssDNA and ssRNA have very different (in general) binding partners. E.g. DNA is interacting with transcription factors, RNA polymerase, histones etc, while RNA is interacting with ribosomes, reverse transcriptase and others.
So, shortly, it is more likely that protein A that binds DNA aptamer will not bind RNA with corresponding sequence.
Also, DNA and this DNA transcript will probably have very different secondary structure.