I believe chepner gets it right in the comment. We can burn lots of things these days. We burn CDs, DVDs; we burn subtitles onto video clips; we burn texts, logos, credits, timecode, and such onto footage; we burn many things onto many things. When we use "burn" this way, the thing we burn will be permanent.
I've just watched this movie a couple weeks ago. The main character and his peer used lots of jargon (as you seem to know in other questions relating to the same movie). It was unclear in what period the setting of the movie was, but I remember that there was a scene that his competitor boasted about his 2.4-megapixel camera, so it could be just a decade back, and they seemed to know how to handle digital stuff well. So, they sure use the word "burn" like IT and hardware people do.
I can't find a good reference for this. I believe that it's jargon used only in an in-group, so unless we have a tech-person who also use this word this way passing by and choosing to answer this question, I believe that it's unlikely that we will have a good reference for this usage.
Having said that, the script actually tells us what he meant. In that scene, he negotiates with Nina (the producer of the news program, if I'm not mistaken), threatening her that he might leave for another network, which has a better offer. Seeing that he has the upper hand, he demands a lot of things from her, as you can see in his long lines (the quotes/subtitles are by MovieQuotesAndMore.com, emphasis added):
Louis Bloom: Now I like you, Nina, and I look forward to our time together, but you have to understand fifteen thousand isn’t all that I want. From here on, starting now, I want my work to be credited by the anchors and on a burn. The name of my company is Video Production News, a professional news-gathering service. That’s how it should be read and that’s how it should be said. I also want to go to the next rung, and meet your team. And the station manager and the director and the anchors, and start developing my own personal relationships. I’d like to start meeting them this morning. You’ll take me around, you’ll introduce me as the owner and President of Video Production News and remind them of some of my many other stories. I’m not done. I also want to stop our discussion over prices. This will save time. So when I say that a particular number is my lowest price, that’s my lowest price. And you can be assured that I arrived at whatever that number is very carefully. Now, when I say that I want these things, I mean that I want them, and I don’t want to have to ask again. And the last thing that I want, Nina, is for you to do the things that I ask you to do when we’re alone together in your apartment. Not like the last time!
[Nina just stares at him]
Louis Bloom: So…I’ll tell you what. I have the van crash on Moorpark tonight. It was a couple of stringers, actually. That could lead by itself on an average night. I’d be willing to throw that in for free. So what do you say, do we have a deal?
To recap the highlighted parts:
I want my work to be credited by the anchors and on a burn. The name of my company is Video Production News, a professional news-gathering service. That’s how it should be read and that’s how it should be said.
The parallel is clearly there (though it's not very neat). In other words...
He wants his company name to be said exactly that by the anchors, and not just that; he also wants his company name to be read that way by the viewers of the news on their TVs; and that's why he says he wants it "on a burn".