What happens when a blood vessel is cut during surgery?
Blood vessels (not just veins, but arteries also) are transected (cut), ligated (tied off), and/or cauterized (burned closed) in just about every surgical procedure ever. When done appropriately, this does not cause a problem because there is more than one path from the heart to the tissue and more than one path from the tissue back to the heart. This is called collateral circulation and often involves vascular anastomoses. In some cases collaterals are quite abundant, for example, the rectal plexus, the vascular supply relevant to the title to your question. Hemorrhoids are, in fact, the result of a very particular kind of venous branching, a porto-systemic communication. The rectal and anal circulation are drained through two parallel pathways, the portal circulation (through the superior rectal vein) and the systemic circulation (through the middle and inferior rectal veins). The portal circulation returns blood from the gastrointestinal system to the liver (and from there to the inferior vena cava and the heart), where the systemic circulation returns blood directly to the inferior vena cava (without passing through the portal system of the liver). When the pressure in the portal system increases (whether through pathological portal hypertension, or other, temporary nonpathological conditions, e.g., pregnancy), blood is shunted from the portal arm through to the systemic arm, engorging the vessels in a few specific areas. The rectal plexus is one of them, and is a clear demonstration of the principle that there is more than one path back to the heart.
An important part of good surgical technique is identifying which blood vessels can be cut, tied, or burned, and which ones can't (or will result in complications if they are).