What is the action ot slowly blowing air on the ignited fire to build heat and intensity called? Intensifying the flame. Making a dying flame alive.
The already mentioned fan is a more precise verb than the following ones are going to be, so consider my answer only supplementary.
To relight (a fire). To set alight or start to burn again.
To feed, stir up, and poke the fire in (a furnace), to tend the furnace of (a boiler). Also, to feed or build up (a fire), and with up.
He blew several bellowsful to stoke the spark before placing a few small pieces of white wood onto the flame and closing the grate. / When he came back he blew on the sparks to stoke up his fire afresh, and warmed himself.
One would fan the fire...
- To move or cause a current of (air) with or as if with a fan.
- To direct a current of air or a breeze upon, especially in order to cool: fan one's face.
- To stir (something) up by or as if by fanning: fanned the flames in the fireplace; a troublemaker who fanned resentment among the staff.
Well I don't have a verb, but I thought it interesting to note that your question describes a device known as a 'bellows'. It is one of those nouns that is plural in form but singular in meaning (though it can be referred to as 'a pair of bellows', see Wikipedia's article). It was invented in medieval Europe and was especially used by blacksmiths to produce streams of air to raise a fire's temperature. It consists of an airtight, compressible chamber fixed between two handles that can be used to squeeze the chamber and empty it of air. There's also a valve that allows air to re-enter the chamber as the handles are pulled apart. So, while it might not be a verb, I guess you could fashion a verb phrase using the word.
"Bellows." Merriam-Webster.com. Merriam-Webster, n.d. Web. 22 Oct. 2013. http://www.merriam-webster.com/dictionary/bellows.