It is called conditional mutation.
You flox (put lox sites around) gene of interest and express Cre recombinase driven by tissue-of-interest-specific promoter.
Illustration from here:
Using chemically-activatable variant of Cre recombinase (cre-ER) you can create knock-out in some cells of tissue of interest, not every.
Addition: a bit weird but still useful method is to produce KO by means of CRISPR/Cas9 system. Tissue-specific expression of Cas9 protein together with gene-specific sgRNA will likely produce KO in mosaic fashion. One of positive sides is that such approach requires only one transgenic line, e.g. Tg(promoter:Cas9;sgRNA)
Addition: (via WYSIWYG's suggestion) you can express shRNA or other gene transcription-interfering product from promoter that only active in specific tissue/specific period of time. And beauty of genetic/bioengineering is that you can combine several methods for additional control, robustness, efficiency etc.
Taken from here:
Inducible and sell-specific gene KO with shRNA
For further reading:
I re-read OP question and, as commented to it, feel some confusion. So I would like to mention as well way to express mutated gene in specific tissue that, when expressed constitutively (all the time, everywhere), is lethal. Imagine some very strong oncogene. Or something like cell-toxic protein, e.g. chemically-activatable nitroreductase or light-activatable Killer-red. Or it might be mutated version of your favourite synaptic protein that, when expressed in all neurons, causes severe lethal phenotype.
What you do, is create transgenic animal with transgene-GFP fusion (nitroreductase, KillerRed, etc) that has floxed stop-codons in front, and cross that animal with transgenic with tissue-specific CreER (for example). What you get is animal that, when injected with tamoxifen, start to express mutated gene/toxic protein in tissue of interest. Then you are sure that only GFP-positive cells express that gene. That is how you "express that mutant gene only in a specific tissue of interest to get a idea of which phenotype it produces".