In other words, what are the disadvantages of playing at the limit of your technical abilities straight away, without warming up first? Why is warming up usually recommended?
Many musicians use a warmup routine that is learned off by heart. They can tell when something feels wrong, but more importantly, it can be used as a very simple way to become ready and get into the groove or flow.
Physically it's the same principle as going to the gym for a workout. If you don't warm up first, you may find your joints are stiff, your movements slower and less precise than you may like, and you may not be able to feel as much as you should.
Warming up can relax tissues, pump oxygen into muscles and tissues, and get you ready for movement.
Not warming up can cause pain and damage - sometimes even permanent - to tissues and joints.
tl;dr - always warm up, even if only for 5 minutes
Why is warming up usually recommended?
To play them well, all instruments (including the voice) require musicians to use their muscles, tendons and nerves in ways not normally used in everyday life.
Because that's so, most musicians, unless they are unusually facile or always ready to play because they play so much (John Coltrane reputedly was almost constantly playing, except when he had to eat or use the toilet - he was probably always "warmed up") , need a warm-up period to stretch and bring better muscle tone, blood circulation and nervous stimulus to those parts of the body used in playing.
Often a performer will do a warm-up set right before a performance. When that entails too much time and trouble, sometimes performers will begin their performance with a "warm up piece" that isn't particularly challenging but allows them to stretch and play in different ways that will be required during the rest of performance.
Aside from actual warm up playing, there are also various sorts of exercises and stretching routines that musicians may do before playing.
Here's some good information about this point: OUR GO-TO ROUTINE: 10 ESSENTIAL STRETCHES FOR MUSICIANS
Many people may not understand how physically demanding playing music can be, especially for those of us who have to hold the weight of an instrument through a long performance (we’re looking at you, marching euphoniums), or who often play lengthy, complex works solo (hello, concert pianists and strings musicians).
Keeping your body in good physical shape is essential to playing your best, no matter your musical instruments. Endurance, control and overall flexibility will aid in preventing common injuries like tendonitis and muscle sprains, improve your posture, and ultimately help you to become a stronger musician. In addition to your regular cardiovascular and strength-building exercises, the following are helpful stretches for musicians to complete prior to every practice and performance. Think of this routine as essential prep prior to playing, just as you’d warm up and tune your instrument...
Neck/Shoulder Movements... Elbow Movements... Wrist and Finger Movements...
It's no different than an athlete going through a warm-up routine before competing. Playing most instruments proficiently is essentially a specialized form of athletic activity, and to be at your best it requires the same sort of preparation/warming up as any other form of athletics.
Baseball players play exhibition games in spring training, take batting practice and warm up in the bullpen before a game, and do stretches between innings and take practice swings with a weighted bat while standing in the on-deck circle. Most serious/professional musicians have similar routines, depending somewhat on the situation, as @LaurencePayne has mentioned in his answer:
What are the disadvantages of not warming up?
Quoting the above page:
Warming up aids in preventing common injuries like tendonitis and muscle sprains, improve your posture, and ultimately help you to become a stronger musician.
So if you don't warm up, you're at risk for those maladies.
It also helps you play better when it's time to perform - your body is ready to go. If you take the stage "cold", your chances of hitting your stride right away are diminished - and that's not good for your performance or your reputation.
I live in NYC. My Dr. shares an office with a physical therapist. I was discussing with him some hand and back problems I was suffering from due to playing (EBG). Doctor told me that most of his office mate's clientèle are professional musicians, who need PT because of the numerous joint, muscle and back problems that arise from their constant work as musicians.
Warming up helps to minimize such problems, but they remain an occupational hazard for many musicians playing at the highest levels of professionalism - full time concert and jazz musicians, for example.
In other words, what are the disadvantages of playing at the limit of your technical abilities straight away, without warming up first?
As far as that goes, the disadvantage is that if you play without warming up, you will not be able to play at the limit of your technical abilities straight away, and if you try to do so, you could injure yourself, just like baseball player can tear his shoulder if he walks in off the street and takes a home run swing at a 100 MPH fast ball.
You could also end up performing poorly, compromising your show and your reputation, as mentioned - nothing worse than opening with your signature riff, but failing to execute it correctly because you're not warmed up.
As a brass player, 'warming up' is partly a literal description - until the instrument is at the temperature of your breath, it won't be in tune! Beyond that, some players seem able to jump straight in to performance mode, others benefit from some 'limbering up' exercises. Personally, I play a few phrases after which I can either say 'that's fine!' and carry on with the gig or 'bit rough today - better play some long notes and lip slurs!'
As a pianist, I find it much the same. Not all gigs are technically demanding. Some days the fingers are 'working well'. On others they feel like a bunch of sausages and some exercises are called for.
Warming up seems to be an individual thing. Some like to warm up before the performance, others design their set list to begin with easier pieces and warm up while actually performing, but either way most will agree the performance feels more comfortable and flows more naturally after they've warmed up. For singers, it helps clear out phlegm and stretches the vocal chords and helps extend range by a step or two, and makes pitch control easier to accomplish. Finger exercises on a fret board or keyboard help loosen hands that may feel stiff. No matter what instrument, warming up can help get mental concentration on the task at hand. There are more advantages I'm sure. The disadvantages are that you may not have these things going for you if you choose not to allow time for warmup.