See https://musicfans.stackexchange.com/a/2421/84 for another angle on this.
The 'warm' sound of vinyl is a misnomer; a better description these days would be 'blurred' or 'distorted'.
When CDs first appeared there was much wailing & gnashing of teeth from the existing audiophile community - to take away our beautiful old valve warmth & replace it with this cold, hard digital noise.
Bits? Who needs bits? We have our multi-thousand pound warm analog hi-fis to listen to. They produced graph after graph & comparison against comparison to prove they were right.
In 1984... they were right. DACs [Digital Analog Converters] had a long way to go in those days. Their ability to subtly shade sound wasn't there yet, they had a coarse edge which people with very good ears could differentiate.
Move on 30 years.
Almost no-one records on tape any more; they record to hard drive via computer & DAW [Digital Audio Workstation] software. Their DACs are no longer mere 16-bit, 44.1KHz approximations of the original sound - now they're capable of 24-bit fixed point or 32-bit floating point at up to 192KHz.
Compared to the frequency response of the human ear, or indeed any monitor speaker ever made, that's complete overkill. Part of the old 'vinyl is best' argument was that frequencies outside of human hearing range were important, even if no-one could actually hear them... & vinyl could go above the 20KHz accepted as the maximum a human can hear.
A recording at 192KHz has a maximum frequency of 96KHz... way beyond what vinyl can capture.
So... take that recording, cut it to vinyl & if you like also crush it down to mp3... then compare.
The original - clean right through, from below to well above human hearing; no background noise, no distortion, nothing but what the engineer put there.
The mp3 - depending on how far it was crushed - slight artefacting at the higher frequencies, an absolute cut-off at about 18k. At higher bit-rates 9 out of 10 people couldn't tell the difference between the original & mp3 played back on a good system. Play it over a regular consumer hifi or any kind of mobile device on headphones & 99 out of 100 couldn't tell the difference.
The vinyl - on 10 grands' worth of kit, using a pristine pressing - will sound very sweet, though there'll still be the rumble of the turntable underlying it all, pops & clicks that seem to get into even brand new pressings; the slight blurring at higher frequencies caused by the cutting lathe & stylus not being at absolutely precisely the same angle as each other; the EQ discrepancies caused by the RIAA encoder & decoder also not being perfectly aligned... & that's before you try it on your standard 500 buck consumer rig with a 2-year old stylus & grooves full of fluff, over randomly chosen components, or worse, some gaudy flashing lights all-in-one 'hifi' where the user can make their own EQ curve...
...these days, don't confuse 'warm' with 'inaccurate'.
In practical terms, if you were to take any record at random, play it on your hifi, record that to your computer over the standard audio inputs, then spit out the result as a CD... it would sound less good than the vinyl did, no matter how good or bad the vinyl was in the first place, because your component spec isn't up in the 'pro' category. The vinyl itself wouldn't sound as good as the mp3 you could buy from iTunes or Amazon etc, for the same reasons.
The only possible reason I could think of for consumer DIY vinyl to digital transcoding would be to rescue vinyl that was no longer available in any other format.