Bence aralarında fark var aslında. Why would i have to lie? = Neden yalan söylemek zorunda olayım ki? Why should i haveo to lie? = Neden yalan söylemek zorundaymışım ki? Çeviriyi anlaşılır yapabildim mi emin değilim ama bence bu şekil bir fark var aralarında :)

Why would I have to lie? - WOULD - This asks what will you have to gain from it. Why should I have to lie? SHOULD - This asks why is it necessary to do it, or what is causing you to do it. They have very close meaning, but WOULD means you have a reason, and SHOULD means something else is causing you to do it.