Yes, under-eye filler can go wrong, and the most serious problems involve the eyes themselves. This page covers what the sources name and which signs to act on.
The risk to know about first
Cleveland Clinic notes that sometimes, hyaluronic acid can be accidentally injected into a blood vessel and this could possibly lead to temporary vision problems.
That is the reason to care about who injects and how carefully they work around the eye.
Signs that something is off
Cleveland Clinic lists extreme pain, loss of vision or seeing floaters at the time of your injection as signs that something could be wrong.
The same source says these are all warning signs that you should go back to your provider and have them look into it.
Where to read the wider evidence
Complication data for hyaluronic acid filler is gathered in a meta-analysis of adverse events reported from facial injections. A systematic literature review of safety and complications with dermal fillers widens the view to filler products in general.
For a consumer checklist, the FDA's page of dermal filler dos and don'ts is short and plain. Nebraska Medicine also explains which filler type suits tear troughs, which matters when you choose a product.
Before and after your injection
Before treatment, ask your provider how they manage the risk near the eye. After treatment, know the warning signs and call at once if one appears.
The minutes during and after injection
The answer changes in the moments during and right after the injection. Pain or vision changes then turn a routine visit into a reason to call your provider.