Assign it to an object and then take a look inside that object with str(). For example:

hmap <- plot_heatmap(...)
str(hmap)

You will likely see a 'slot' (variable) that contains the gene names and / or their cluster assignments.

Try saving it to a large pdf or as an image.

pdf("heat maps.pdf", height =1000, width= 1000)

plot_heatmap{your code}

dev.off()

This should save the plot to your working directory. You can also change pdf to png for an image.