"Stress" in English is a bit complicated. In one sense, both words in a phrase like "warm up" are stressed.
One way of defining "stress" is in terms of whether the vowel in a syllable can be reduced. If the vowel can be reduced, the syllable is certainly unstressed. If it can't be reduced, the situation is not so clear. In a phrase like "warm up", normally neither syllable can be pronounced with a reduced vowel: it can be transcribed phonologically as /wɔrm ʌp/ (or /wɔːm ʌp/ in a non-rhotic accent). (In fact, I'm simplifying the situation here: it actually is possible to hear phonetic reduction of vowels in contexts where the phonological rules that I'm using would say that vowel reduction should not be possible. But for a learner, I think it's useful to stick to the simplification that we only see reduced vowels in unstressed syllables.)
As an independent word, warm is definitely stressed, and we can say that it doesn't actually lose that stress in the phrase warm up.
To describe what happens, we can introduce a distinct concept of "accent". I said that stress can (in simplified terms) be defined as being about whether a vowel can be reduced; accent can be defined as being about whether a syllable takes high pitch (or alternately, we can say that there is a pitch fall after an accented syllable in English). (As with stress, my description of accent here is a simplification: you can find a more accurate description with some more details in the paper "A Theory of Pitch Accent in English", by Dwight L. Bolinger, 1958. It might be a bit more accurate for me to say that an "accented" syllable tends to be marked by a change in pitch: often either a rise in pitch before it or a fall in pitch after it).
Many types of words or phrases have multiple stressed syllables, but are accented only on the last stressed syllable. The accented syllable in a word is sometimes said to have "primary stress" (written as ˈ before the syllable) and stressed syllables in a word before the accented syllable are said to have "secondary stress" (written as ˌ). But in fact, in special situations, the accent can sometimes be placed on one of the earlier stressed syllables in a word or phrase.
Two situations that commonly cause a word to be accented on the earlier syllable are emphasis (e.g. showing a contrast), or avoiding a sequence of two accented syllables in a row.
For example, the word Chinese, which has two stressed syllables, is normally pronounced with an accent on the second: /ˌtʃaɪˈniz/. But it can be accented on the first syllable to emphasize a contrast ("I said that they were Chinese /ˈtʃaɪniz/, not Japanese /ˈdʒæpəniz/") or when it is used as an attributive adjective before a word that starts with a stressed syllable ("a Chinese city /ˈtʃaɪniz ˈsɪti/").
So in theory, warm up could be accented on the first syllable to emphasize a contrast ("take your time to ramp up and warm up /ˈræmpʌpn̩ˈwɔrmʌp/") or when it is used attributively before a word starting with a stressed syllable (in that context, it would be written with a hyphen, as in "a warm-up exercise /ˈwɔrmʌp ˈɛksərsaɪz/").
But your example doesn't fit either of those contexts. And in fact, to my ears, the speaker saying "Put the fire on. I need to warm up" puts the accent on up, just as would be expected from the usual rule! Compare the pitch contour of fire on earlier in the sentence, where fire is accented and on is not, to the pitch contour of warm up: I think there's much less of a drop in pitch between warm and up than there is between fire and on. I might be wrong (I haven't analyzed the pitch with any audio tools), but I think you might be hearing the use of a "stressed", as in unreduced, vowel in warm and misinterpreting that as "primary stress", when in fact the primary stress/accent is on the syllable up.
