Chorophyll and photosynthesis in plants with coloured leaves
Most of ornamental (often variegated ) plants do performs photosynthesis. They do have chlorophyll even if their color is not green. They have non-green color due to various different pigments. These extra color has their own functions like blocking harsh sunlight or protection from insects (Karageorgou and Manetas 2006). 

In many non-green plants carotenoid content is way higher than chlorophyll. One of the main functions of these carotenoids is to expand the range of wavelength where photosynthesis can happen (Hashimoto et al 2016). This is especially important for such plants living in low light conditions or other stress factors are dominant. 

Some comparison of content in different color leafs of Acer saccharum is enlisted in table below. Image is taken from (Junker and Ensminger 2016)



Photosynthetic activity of such plants can be measured by chlorophyll fluorescence techniques (Borek et al 2016). Photosynthesis performance of many of these plants (like Actinidia kolomikta) is actually properly maintained (Wang et al 2016) and does not have high Photosynthetic Cost.

Just to extend this answer for little general information, plants without any chlorophyll (like Cuscuta europaea) are often parasitic plants and they need host to survive and get nutrients.