Instead of using layers with selected features (geology and vegetation), export selected features into separate shapefiles, and use these two separate shapefile as input data of intersect operation. To do this,

1. While relevant features are selected from geology (i.e., lime stone), right-click on the layer Geology, Data, and Export Data. Save selected features as shapefile named geo_sel

2. 1. While relevant features are selected from vegetation (i.e., dense conifer), right-click on the layer Vegetation, Data, and Export Data. Save selected features as shapefile named veg_sel

3. Open Intersect tool, and add geo_sel and veg_sel for input data, instead of Geology and Vegetation

Make sure you save the output feature class as shapefile.