Is there a way to make 0 values a different color in a MatrixPlot?

SeedRandom[1]
outtabletoprint = RandomInteger[{0, 5}, {20, 20}];

1. You can replace 0s in the input matrix with Black:

MatrixPlot[outtabletoprint /. 0 -> Black, 
 ColorFunction -> "TemperatureMap", PlotLegends -> Automatic]

2. You can use the option ColorRules:

MatrixPlot[outtabletoprint, ColorRules -> {0 -> Black}, 
 ColorFunction -> "TemperatureMap", PlotLegends -> Automatic]

same picture

Note: Both methods work in version 9.0.1 (Windows 10 - 64 bit), in version 11.3 (Windows 10 -64 bit) and in version 12.1 (Wolfram Cloud).

color[z_] := Which[z == 0, Black, 0 < z <= 1, ColorData["TemperatureMap"][Rescale[z, {0, 1}]]]

Legended[MatrixPlot[outtabletoprint2, ColorFunction -> color,  ColorFunctionScaling -> False], 
 BarLegend[{"TemperatureMap", {0, 1}}]]

Or

MatrixPlot[outtabletoprint2, ColorFunction -> color, 
 ColorFunctionScaling -> False, 
 PlotLegends -> BarLegend[{"TemperatureMap", {0, 1}}]]

If you really want Yellow-Red color, you can use SolarColors in reverse order.

color[z_] := Which[z == 0, Black, 0 < z <= 1, ColorData[{"SolarColors", "Reverse"}][Rescale[z, {0, 1}]]]

MatrixPlot[outtabletoprint2, ColorFunction -> color, 
 ColorFunctionScaling -> False, 
 PlotLegends -> BarLegend[{{"SolarColors", "Reverse"}, {0, 1}}]]