You could convert your elevation data to a vector (Raster to Polygon) then run an intersect against the vector elevation and the polyline tracks.
You can dissolve the output by the original FID of your tracks whilst applying a "min" statistics on the field that contains the elevation value (that was incorporated during the intersect).
This should then return you your tracks data with the minimum height of the track. The min statistic can be swapped out for for a different statistic if desirable but if you are interested in pulling out tracks over a certain height then applying a min statistic during the dissolve will show you the lowest point of that track.
hope this helps.
Ross