Symptoms
Errors: 'Your browser's compatibility mode is not supported. Your browser is supported but the document mode set by the hosting page is not." and "Your browser is not supported. K2 smartforms is not supported with your current browser. Please download the latest version of one of the following browsers"
Diagnoses
Document mode version 8 was being forced on SharePoint. Likely a group policy forcing domain-specific compatibility settings. Changing the document mode to a supported browser version allowed the form-viewer web part to load the form.
Resolution:
Some options are to change SharePoint's master page compatibility meta tag to IE=11 instead of IE=8:
<head>
...
<meta http-equiv="X-UA-Compatible" content="IE=11">
...
</head>
Note that this should be done specifically to pages that house K2 Forms as changing the document mode in SP 2010 can make some ribbon controls misbehave.
Additionally, you may be able to change the document mode via IIS, specifically for the K2 Runtime site, by modifying the 'X-UA-Compatible' HTTP Response Header on the K2 - Runtime site in IIS. The field should have values like the following: "IE=11,IE=10,IE=9,IE=8". Removing the lower versions would force Internet Explorer into a modern Document Mode for Runtime SmartForms, which may resolve this issue.