When you fill fields in the TD manager there is no spell checker. Many fields and switching between them quickly. I rely on the fact that you can trust spell check provided by the browser. Because of typos might be accused of vandalism. When editing a template in the article in visual editor, in the fields spell checking works. Spell check should work in the TD manager too.
Firefox 41.0
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Description
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WFM. The inputs are standard textarea and text inputs, without the spellchecker disabled.
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This is about the TemplateData GUI editor, but the point stands for the fields in the template dialog too.
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Chrome only seems to start doing spell checking in the TemplateData GUI editor inputs for me when I make a linebreak in a textarea, not before that and not in the normal text inputs.
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WFM in Chrome & Firefox. Again we aren't doing anything special, so this would be a browser bug.