Why are desktop folders suddenly showing file dates?

1. The Issue:

You have perhaps inadvertently made a change to the Nautilus settings relating to icons, their 'Zoom' level and/or the default information that is displayed underneath them.

Note that the amount of 'Zoom' applied to the icons will also affect the total information seen below the icon and there are 3 available information settings, each of which can be seen at different Zoom levels.

2. The Solution:

To rectify the issue you have seen follow this trail from within a Nautilus window:

Edit --> Preferences --> Display --> Icon Captions

The setting to look for is the 'Modified' tab at the top of the 3 selection boxes which I believe you will find has been selected. Below is a screenshot to demonstrate where I have repeated your setting and demonstrated the appropriate dialogue box to alter this setting:

Experiment a little with these settings (if the top box is found 'Modified' set to None') and all should then be well. Note that for the Desktop a refresh will be required, perhaps the easiest way is to simply log out and then log back in...

2b. Altering Zoom:

If you simply wish to alter the 'Zoom' value of your Desktop icons (making these icons smaller will doubtless also conceal the Modification times) follow a similar trail from within a Nautilus window:

Edit --> Preferences --> Views --> Icon View Defaults

and alter the Zoom level until you are happy. A screenshot from my own system:

And then all should be well :)