This interaction relates to a broad family of hormones called catecholamines. There are many examples and they share a common structure.
Notice the common structure of catechol (the carbon ring with two hydroxyl (OH) groups) and an amine (NH2) group.
This group of hormones and neurotransmitters act as part of the sympathetic nervous system, which is one of two mutually-antagonising arms of the autonomic nervous system. Both are active all the time to maintain homeostasis, but the sympathetic nervous system also stimulates fight or flight activity (increase heart rate, blood pressure and cardiac output, for example), while the opposing parasympathetic nervous system promotes more routine tasks (rest and digest, feed and breed).
Methamphetamine is a stimulant drug. This information is taken from a paper on the pharmacologic mechanisms of crystal meth:
Methamphetamine causes the release of the neurotransmitters dopamine,
norepinephrine and serotonin and activates the cardiovascular and
central nervous systems.
Methamphetamine is similar to other amphetamines and cocaine, in that it has stimulant properties, but it also has a longer elimination half-life. It takes longer for it to be broken down and removed by the body.
Dental anaesthetic usually has two components; an anaesthetic (often lidocaine) and adrenaline (epinephrine) which causes localised vasoconstriction to reduce bleeding.
Lidocaine can have adverse effects if a significant amount is absorbed systemically. The effects tend to depress the cardiovascular system, as opposed to the stimulant effect of catecholamines. However, it is the adrenaline which confers the risk in this case, as it is also a catecholamine.
The problem arises due to excessive stimulation of the sympathetic nervous system due to the increased levels of these catecholamines as a result of methamphetamine use, combined with the absorption of additional adrenaline as part of the dental anaesthetic.
This can result in haemodynamic instability and heart arrhythmias (like ventricular tachycardia) as part of a catecholamine surge.
A local anaesthetic without the vasoconstrictor can be offered if there is concern about the risk.
There are some diseases in which there is increased endogenous catecholamines (such as pheochromocytoma). Dental anaesthetic could have a similar risk in these cases. I cannot find specific evidence fo quantify the risk, but this may reflect the fact that these conditions are rare.
Even in common conditions like ischaemic heart disease and hypertension, the vasoconstricting adrenaline may be avoided by dentists.
To answer your final question regarding the stimulant medication used to treat attention deficit hyperactivity disorder (ADHD), I found a source of professional dental education.
Local anesthesia should be used with caution with the patient with
ADHD. The dental practitioner should seek to achieve profound
anesthesia so that endogenous epinephrine is not released, which could
react with medications (especially methylphenidate). In addition, an
aspirating syringe must be used to avoid intravascular injections and
the possibility of a summation of drug effects from the
vasoconstrictor, which could increase BP and heart rate.
So it seems there is a risk with methylphenidate, but this paper implies that it is not as significant as might be seen with methamphetamine use. This is perhaps not surprising as licensed medication will be precisely dosed and monitored, unlike unregulated illicit drugs. Note the suggested use of an aspirating syringe to ensure the anaesthetic mix is not injected intravascularly.
It is also interesting to note the concept of catecholamine release simply due to pain. This demonstrates why good local anaesthesia is useful in the first place.
Local anaesthetics and medically complex patients
Pharmacologic mechanisms of crystal meth
Dental implications of the patient with ADHD