First, Neandertals could have interbred with anatomically modern humans and passed DNA on to us, without leaving a trace in mitochondrial DNA. As has been said repeatedly, different parts of the genome have different MRCAs, and the mtDNA MRCA could easily have been a H. sapiens while the MRCA for some other part of the genome was in the common ancestor of sapiens and Neandertals. Put another way, even if Neandertals contributed mtDNA to sapiens, that contribution could have subsequently disappeared by genetic drift.
Second, all we can say at present is that Neandertal mtDNA is either extremely rare or absent in modern humans (since many thousands of mtDNA samples have been studied to date). It is possible in principle for Neandertal mtDNA to have persisted in modern humans (modern Europeans, to be specific) at a very low level, but it is unlikely. The population Neandertals would have admixed with would have been small (in the thousands, probably). If Neandertal mtDNA were at a low frequency, say 0.1%, then it would have only existed in a handful of copies; the odds are that it would either have increased in frequency or (more likely) disappeared entirely. I don't know how many European mtDNA samples have been studied, but it must be in the thousands, so any variants as common as 0.1% would probably have been seen by now.
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