The big issue revolving around HHS is Obamacare. So it makes sense that Obamacare will be the reason for opposition to Dr. Price. To find this look no further than partisan politics.
Republicans want to repeal Obamacare, while Democrats want to keep it. Politically, the Democrats want to portray Republicans as destroying the health care system and leaving millions uninsured. As a member of the House of Representatives, Tom Price has introduced alternatives to Obamacare ever since the bill was passed (and probably before).
So here is the reason Democrats oppose Tom Price so strongly. He might propose a working alternative to Obamacare and leave them having to defend a system which isn't overly popular to begin with. Democrats either need Obamacare to stay in place or to have it replaced by obviously worse.