It is easy to show that Paul did not write the Epistle to the Hebrews, in fact that is the view of almost all modern scholars, who generally do not regard Hebrews as an epistle at all. Although attributed to Paul quite early, even many of the Church Fathers expressed doubts about Pauline authorship. When considering other possible authors, Luke was not among those they considered the main contenders.
The possibility of Lukan authorship depends in part on the reference to "our brother Timothy has been set free" in Hebrews 13:23. Interestingly, the King James Bible subscript attributes the letter to Timothy, in spite of this verse. In any event, only the closing verses seem to identify the book as an epistle, so they need not be original.
The hypothesis of authorship by Luke also depends on the assumption that Luke, the physician and companion of Paul, wrote the gospel that now bears his name. The gospel was originally anonymous and was only attributed to Luke later in the second century, so that most New Testament scholars now doubt that he really was the author of the Gospel of Luke. If Luke did not write the gospel, then there is no reason to believe that he wrote any other book in the New Testament.
This still leaves open the possibility that 'Luke', the actual author of Luke's Gospel was the author of Hebrews. A difficulty for any such hypothesis is that it is almost universally accepted that Hebrews was written prior to the destruction of the Temple in 70 CE, whereas scholars now say that Luke was written no earlier than the 90s of the first century. Also against this view is the fact that Luke/Acts was written with a strong emphasis on gentiles and is assumed to have been written by a gentile, whereas this does not seem the be the case with Hebrews. By comparison, there is nothing to support the option of Hebrews having been written by the same author.