The Peek Templates can be shown either on mouseover (see the ShowPeekTemplateOnMouseOver property), or automatically on a timed interval (see the ShowInterval property): http://www.telerik.com/help/aspnet-ajax/tilelist-tiles-peek-template.html.
Since touch devices do not have a mouseover event, the first option can hardly be used on them, but there is no automated way to get this to happen out of the box, because using the tap (click) will override the other functionality available when a tile is clicked: http://www.telerik.com/help/aspnet-ajax/tilelist-client-side-events-onclienttileclicking.html.
What you can do is to use the OnClientTileClicking event to show the peek template via the tile's API (see the showPeekTemplate() method): http://www.telerik.com/help/aspnet-ajax/tilelist-client-side-tile-api.html. You may also want to cancel it in this case, depending on the rest of the functionality you have.
You can also add a check for the $telerik.isTouchDevice flag from our static client library (http://www.telerik.com/help/aspnet-ajax/telerik-static-client-library.html) to execute this logic only for mobile devices.
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