I would prefer not to wear a linen tie with a linen suit. It could be done, sure. But generally speaking, I don't think it tends to be the best combination.
With a linen suit, there's actually nothing fundamentally wrong with the tried and true silk necktie. The textures tend to be different enough to keep things interesting, and I don't associate silk with winter, the way I sort of associate cashmere with winter.
A more definite recommendation of a good "summer tie" would depend on factors you haven't mentioned, such as the color of the suit, your geographic location, the setting where the suit and tie will be worn, your choice of shirt (collar, color, and so forth), your coloring, etc. Although, of course, certain colors are typically associated more strongly with summer, than they are with winter, and this could influence one's choice of summer tie, as well.
BTW, I recently purchased a Brooks Brothers navy blazer, that's 37% silk, 33% linen, and 30% wool. With that sucker, I figure I'm ready for any type of tie and any season of the year.