I noticed with waterless a/b extractions that if the basing reaction proceeded too fast, or started getting out of hand, yield in pseudo went down. I learned by experience that it was far better to do a slow basing with dry alcohol than it was to use a little water, since water seemed to make the process heat too much and yields were always off when this happened. Even with no water or very little water being used, heat is generated; holding the beaker in your hand as you stir the pill mass in a waterless a/b will show that the freebasing reaction itself is exothermic, although not so exothermic as the disassociation of NaOH in water.
I do not think the "burning" of pseudo is the result of the freebasing of the pseudo itself, divorced from the disassociation of NaOH in water. I don't even know that what happens can be called "burning." Pseudo is pretty tough molecule. In dry environment the rapid heating of NaOH caused by a little moisture may be too much. I doubt this is due to the freebasing of the pseudo molecule itself. I was seeking in the waterless a/b method to avoid releasing water activated foilants in the pill mass which made using a NaOH solution impractical.
The use of a cooled aqueous solution of NaOH should not harm the pseudo. It may activate any water-activated foilants present, however, and make a mess of the pill mass.