My partner deactivated his Facebook account. Yet when I went on Facebook his name/pic had a green dot next to it. Is he lying?

I wouldn't assume he is lying. I think it's totally possible that he is telling the truth.

The last active time seen on the facebook app and/or the green dots aren't accurate. It's possible he may have deactivated his facebook account, but is still logged into a Facebook app on his phone or just the general Facebook log in on an iPhone (in Settings), which is some how triggering the Green dot to show up, even though his account is deactivated. Either way, I wouldn't put too much stock into what you see on the Facebook app and assume he is active (or possibly going behind your back and talking to some other woman).

Until he gives you a good reason not to believe him, I would take him at his word for the time being.