The significance of the near-verbatim agreement is evident if we look at the three texts in question together:
Matthew 10:10 ἄξιος γὰρ ὁ ἐργάτης τῆς τροφῆς αὐτοῦThe only difference between Luke 10:7 and 1 Timothy 5:18 is that the latter has dispensed with γὰρ ("for"), and this is easily explicable given the syntax of 1 Timothy. Between Matthew 10:10 and 1 Timothy 5:18 there is this variant and one other: where Luke and Timothy both have τοῦ μισθοῦ αὐτοῦ ("their pay"), Matthew has τῆς τροφῆς αὐτοῦ ("their food"). This reduces but does not exclude the probability that 1 Timothy is citing Matthew's 10:10, while the degree of verbatim agreement is such that a citation of Luke 10:7 seems highly probable. Indeed, virtually any argument that would not imagine that 1 Timothy 5:18 is citing either Luke or Matthew would have to invoke a hypothetical entity as the source of this material. And such invocation always gets tricky.
Luke 10:7 ἄξιος γὰρ ὁ ἐργάτης τοῦ μισθοῦ αὐτοῦ
1 Timothy 5:18 ἄξιος ὁ ἐργάτης τοῦ μισθοῦ αὐτοῦ
As I noted in my discussion of Robinson's treatment of 1 Timothy in Redating the New Testament, he completely fails in that context to address this datum, even though it stands as strong counter-evidence to his decision to date 1 Timothy two to four years earlier than Luke's Gospel. He does however address it in passing, in a footnote on p. 183, a full hundred pages after his treatment of 1 Timothy on pp. 82-84. The reason for his treatment of the datum at this point, in the midst of his discussion of 2 Peter and Jude, need not concern us here. More interesting is what he says in that footnote. Robinson tells us that "'The labourer is worthy of his hire' could well be a proverbial saying, not a quotation from Jesus." Here we see exactly what I predicted was necessary above: the invocation of a hypothetical entity as the source of the material. But any time such hypothetical entities are invoked, the question of necessity is raised: do we need to posit the existence of such an entity? If not, if we can account for the data without such an entity, then the principle of parsimony takes over and we can dispense with the hypothetical entity. That seems to be the case here. While it is certainly the case that the phrase found in Luke 10:7 and closely paralleled in Matthew 10:10 could have come from proverbial wisdom, either instead of from Jesus or because Jesus spoke a preexisting proverb, the fact that 1 Timothy speaks in terms of a writing should incline us away from thinking that the author of the epistle (i.e. Paul or Pseudo-Paul) derived it from such a source. And when we have ready-at-hand two extant candidates for possible written sources of this material, we should only invoke hypothetical source texts when those two candidates have been clearly excluded. The mere possibility that 1 Timothy is citing a no-longer-extant hypothetical source does not constitute such exclusion.
It is actually quite strange that Robinson feels the need to posit this particular hypothetical source, i.e. a vaguely defined "proverb." He's already argued that there was a proto-Matthew as early as 40, whereas he dates 1 Timothy to 55. It wouldn't have taken much for him to say "Paul is citing proto-Matthew here, and representing a variant in the tradition that also shows up in Luke's Gospel a couple years later." As I find his argument for proto-Matthew less-than-compelling, I would still take issue with such a narrative, but at least it would reduce the number of hypothetical entities by one; such reduction is to be welcomed. As I stated in my discussion of Robinson on the date of 1 Timothy, and will state again, his failure to deal adequately with 5:18 greatly mars his treatment of that text.