Can your venous dorsal arches in your feet be removed if they're reticular and over-dilated? (Photos)

Those are superficial veins and can be treated without any physiologic issues. They happen to be too large for sclerotherapy to be effective. You need microphlebectomy. You can send me a message on the link below for cost in the office setting. This will save you a LARGE out of pocket fee from an ambulatory care center (facility fee, anesthesiologist fee, etc). Dr KaramanoukianCertified vein specialist with ABVLM credentials

Consider first a full venous reflux evaluation with a vein specialist to determine if an underlying venous insufficiency is present in your lower extremities. Healthy leg veins contain valves that open and close to help the blood return back to the heart. Using ultrasound to scan the lower extremities, the vein specialist will determine if venous insufficiency or reflux is present. Signs and symptoms of venous reflux are many including varicose veins, leg or ankle swelling, leg heaviness and fatigue, leg pain, aching or cramping, burning or itching of the skin among other symptoms. Your physician can help you make an informed health care decision on what treatment or combination of treatments is best for you according to your scan results and specific goals.
Best, 

If the cause of large veins is incompetent valves, tested by venous dopler, then treatment is indicated.If there is no valve disease there is no reason to remove normal veins, that help in blood return. You will run the risk of prolonged swelling

Dorsal foot veins can be treated.  You will not lose your foot from treating these veins but there is a chance of some swelling which is temporary.  Larger veins can be treated by microphlebectomies and smaller by sclerotherapy.  If veins treated by sclerotherapy get hard then they can easily be evacuated with a small needle.