The only burn to date was from a soap ingredient (cinnamon in my case). I avoid anything that lists cinnamon as an ingredient now and haven't had an issue since.
Yes, sounds like you are experiencing razor burn indeed.
It has happened to me, the lather quality wasn't good, and the first pass was a little harsh on the skin. When I reapplied the lather, I felt a burning sensation. At the end of the shaving the skin was smooth, no redness, but I felt my face a little sore for a few hours.
The only three ingredient that cause my skin irritation are Almond, Cinnamon and Menthol, it's very important to keep a track record of which soap ingredient that doesn't agree with your skin.
Only a couple over the years have bothered me. L&L Original and Aqua di Parma. I also learned long ago that menthol/eucalyptus potentially can irritate my skin...
I get a bit of a burning sensation from PannaCrema Pure2O Vetiver. I've never had it happen with any other cream or soap, so at least it seems to be a pretty isolated thing.
I've not had burn, but I've definitely had an uncomfortable tingle from some soaps (as soon as it is applied, before the razor touches the face, so it was definitely the soap in those cases). Cella in particular, but also Vitos. I assume it is something in the almond fragrance that disagrees with me. Leaving the soap uncovered and allowing the fragrance to dissipate fixes this issue for me. Fine with Vitos; Cella gets an unpleasant discoloration (turns brown) that doesn't seem to affect performance or negatively impact aroma.
I used to get burning sensations from soap. It was much reduced when using Mitchell's Wool Fat, but it was still there. I never really gave it any attention though. Subconciously, I guess I assumed it was normal. That's until I used Soap Commander for the first time. See my avatar. Now, I'm not trying to push Soap Commander as there are many other artisan soaps out there that probably do the same thing, especially the unscented varieties. However, the first time I used Soap Commander I was amazed that I didn't have any irritation, which is what I thought I was calling a "burning sensation". It wasn't irritation from the shave, it was a burning sensation from the soap. The Soap Commander left me with no irritation or burning sensation.
So, my suggestion is to try an artisan soap that is marketed as hypo alergenic or unscented, all natural....like Soap Commander, or another brand that is marketed for the same purpose.
The current soap I'm using does that to me, it's a slight burning sensation without any visible signs.
Probably I'm sensitive to one of the ingredients..
I once heard that a brush too scratchy can cause that too.
Another possible source of "burning"... On a second (or third, or...) pass, the soap may sting a bit if the first pass was a little rough, and the skin was scraped a bit, or if there are weepers.
Catie's Bubbles LPV irritated my face terribly. I had to get rid of it which is unfortunate as the scent was awesome. Also Reef Point dragon's blood, but that was no big loss as I couldn't stand the scent.
I've always had an allergic reaction to Crabtree & Evelyn Sandalwood hard soap. I get massive razor burn and inflammation after every use of this soap.