What is the origin of the Buddhist expression "Practice as if your hair is on fire"?

I am not sure if this information is entirely appropriate to your question, however there is a meditation technique that came to mind when reading your question and the equivalent sutras.

No idea what the official name for it is but it's basically burning of energy through activating the nervous system - the shivers or goosebumps. Basically, I activate the sensation of goosebumps (yes, just with a thought - however it can still activate with a elevated emotional stimulation) and if you can associate the sensation of getting those shivers all over and often it feels like it comes off your head and shoulders like fire.

I practice multiple times daily and lately the notion has been arising to observe what the limit would be - as I have been observing this technique diligently for a few years and have reached a bottleneck, so to speak.

This research paper may give a little insight; its a real thing. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6071615/

!! A heads up; anyone who gets any ideas about trying it for yourselves I will express very clearly that energy has to come from somewhere in the body. If you burn yourself out too strong and too often, it can potentially be taking energy away from vital processes - Therefore exercising caution is essential.

For those who are already "voluntary piloerectors", the methodology is simple:

• Get control of activation, turning it on when you want to.
• Hold it for as long as possible, which leads to pressure.
• Isolate anywhere in the body that it seems to get blocked and clear those blockages.
• Circulate it long enough to keep it going in a continuous stream.

For those who are not; do not stress yourselves too much about it. Our physical constitution can potentially have an impact - I know people who can't remember ever getting goosebumps - if you can though, good here are some steps to gather control over voluntary piloerection:

• Familiarise yourself to the sensation, associate with things that stimulate your emotions to recreate this reaction (music, sporting events, sentimental situations).
• In the case of music, make a Goosebump Playlist or something of the sort.
• When you notice getting goosebumps, pay attention to where they come from inside, because they have to start from somewhere.

From my personal research, I hold the hypothesis there is a correlation between this sensation of "hair like fire" goosebumps, the hypothalamus (the seat of circadian biorhythms, emotion, hunger and thirst, growth, etc) and the condensed mitochondria in the pockets of brown adipose tissue. The reason why is because I burn off energy, I am less stressed as a result, because I don't have the energy to stress after 'burning it all off". To validate that hypothesis is something to find out when I am far enough in my degree to research it 'officially' haha! Apologies for the digression.

Cultivate in harmony

Samvega is the very root of practice.

Affirming the Truths of the Heart: The Buddhist Teachings on Samvega & Pasada, by Thanissaro Bhikkhu (1997; 5pp./14KB) Popular interpretations of Buddhism today often ignore the importance of two powerful emotions, emotions that propelled the Buddha — and all those who have sought Awakening since — towards the goal of Awakening: samvega, a sense of urgency to escape the round of meaningless existence; and pasada, a clarity and serene confidence that allows one to proceed confidently towards the goal without lapsing into despair. In this short essay the author explores the meaning of these essential emotions and how we can encourage them to blossom in our lives.

Verbal mentions are very often, since an idea of Dukkha, and the urgency to escape, are the righteous reasons to practice the Eightfold Path: Person with Turban or head on fire: AN 6.20 (a, b), AN 10.51, AN 10.54, Thag 1.39 (already generous mentioned by Upasaka Dharmasena here) and point out a very prerequiste to practice: aroused persistence:

(Note: this answer has not been given with the agreement to be means of trade or the purpose of trade and keep people trapped and bound. How you handle it lies in your sphere, but does not excuse the deed here either.)

The Blessed One:

“Short is the life span of human beings, The good man should disdain it. One should live like one with head aflame: There is no avoiding Death’s arrival.”

Source : SN 4.9 Paṭhamaāyusutta https://suttacentral.net/sn4.9/en/bodhi

Its referred to in the Phena Sutta (or Phena Pinduma Sutta) as follows, translated by Thanissaro Bhikkhu:

Thus a monk, persistence aroused,
should view the aggregates
by day & by night,
mindful, alert;
should discard all fetters;
should make himself
his own refuge;
should live as if
his head were on fire —
in hopes of the state
with no falling away.

Alternative translation here by Bhikkhu Sujato:

An energetic mendicant
should examine the aggregates like this,
with situational awareness and mindfulness
whether by day or by night.

They should give up all fetters,
and make a refuge for themselves.
They should live as though their head was on fire,
aspiring to the imperishable state.”

Tissa (Thag 1.39) {Thag 39}

As if struck by a sword, as if his head were on fire, a monk should live the wandering life — mindful — for the abandoning of sensual passion.

Source: http://www.accesstoinsight.org/tipitaka/kn/thag/thag.01.00x.than.html#passage-39