No, I was just wondering while watching various titrations on Youtube. I'm self studying for AP Chem.
According to me, it shouldn't because in the equation N1V1 = N2V2, all I need are 3 knowns to determine the one unknown. And I will get the volume of both the solutions regardless of whether I take the analyte or titrant in either.
Some strong bases can damage the burette if you are leaving them in for extended periods. Also your choice of indicator can be a factor as the color change can be more easily detected one way or the other. If you are doing two or more titrations with an acid or base in common the choice of what will be in the burette will be one of convenience.