How does hyperthyroidism cause diarrhea?
Hyperthyroidism leads to an overactive sympathetic nervous system, hypermotility and diarrhea (Culp and Piziak 1986) (Geffner and Hershman 1992) (Maitra 2015, p. 1084) (Thomas et al. 1973). A demonstration of this is the patient presenting with the sole symptom of intractable diarrhea. It was found to be secondary to hyperthyroidism, and was treated successfully with β-adrenergic antagonists (Bricker et al. 2001). 

Was the hypermotility due to excessive β-adrenergic activity, or insufficient α-adrenergic activity?

An article by Silva and Bianco (2008) states that β-receptor expression is under transcriptional control of a thyroid response element, and is upregulated by increased thyroid signalling. There is much less research on α-receptor expression, I would be much obliged if anyone can provide me with access to the following article:


  *Bilezikian, JP, Loeb JN 1983, ‘The Influence of Hyperthyroidism and Hypothyroidism on α and β-Adrenergic Receptor Systems and Adrenergic
  Responsiveness’, Endocrine Reviews, vol. 4, no. 4, pp. 378-388.*


In any case, they add that increased β-receptor expression is rather modest, often overestimated, and cannot account for the magnitude of amplification in responsiveness induced by thyroid hormone (TH). Rather, thyroid signalling increases cAMP levels associated with a2 and β-receptors to a degree which could account for these changes. 

Increased cAMP occurs during normal β-receptor function but is significantly potentiated by TH in several ways: increases in adenylyl cyclase; downregulation of β-receptor G-protein subunits, Gai and Gβ; and downregulation of phosphodiesterases which degrade cAMP. This explains the effectiveness of the β-adrenergic antagonist noted above, as propranolol decreases cAMP levels, thus decreasing hypersensitivity of the hyperproliferated β-receptor (Silva and Bianco 2008).


  “The reduction of the sympathetic activity associated with
  hyperthyroidism outflow (via increased rate of norepinephrine
  metabolism to decrease overall levels) is likely overridden by the
  increased sensitivity, as suggested by the clinically significant
  amelioration of the adrenergic symptoms with β-adrenergic blockage and
  the consistent observation of elevated cAMP production.” (Silva and
  Bianco 2008).


What are the effects of preferential/hyperactive β-receptor function on gastrointestinal motility?

The β1-receptor increases ghrelin secretion from the stomach – a peptide secreted when the stomach is empty to increase hunger, increase gastric acid secretion and gastrointestinal motility. The β2-receptor promotes smooth muscle relaxation (decreased GI motility) but contracts sphincters of the GI tract (Adrenergic receptor 2018). 

β-adrenergics also promote the secretion of gastrin – a peptide which increases antral muscle mobility and promotes stomach contractions; relaxes the pyloric sphincter, which increases the rate of gastric emptying; is involved in the relaxation of the ileocecal valve; induces pancreatic secretions and gallbladder emptying; and contributes to the gastrocolic reflex (Gastrin 2018) (Katzung and Trevor 2014, p. 143-144). 

What are the effects of relative α-receptor deficiency on gastrointestinal motility?

Considering that β-receptor antagonism causes a relative increase in α-receptor activation we should consider them also. The α1-receptor causes smooth muscle contraction. The α2-receptor causes negative feedback on norepinephrine signalling, relaxes the gastrointestinal tract when activated on presynaptic terminals and contracts of sphincters in the GI tract. It does this by decreasing levels of cAMP (whereas β-receptors work by increasing cAMP) and so it will become less active as TH becomes more active (Adrenergic receptor 2018) (Silva and Bianco 2008).


  What are the adrenergic-mediated effects of hyperthyroidism on gastrointestinal motility?
  
  Under prolonged conditions of excess TH overall peristalsis will
  increase due to mixed effects on smooth muscle contraction primarily
  via α1, and β2. There will be an increase in hunger and gastric acid
  secretion via ghrelin (Adrenergic receptor 2018). Hyperphagia and 
  hypermotility are both suggested to play a part in steatorrhea 
  (Culp and Piziak 1986).


How does hypermotility cause diarrhoea?

In addition to the mechanisms above, it has also been suggested that excessive fatty food intake contributes to the steatorrhea seen in patients with hyperthyroidism. This is compounded by a reduction in time that digestive secretions can act on chyme, ultimately leading to a reduction in absorption and an increased liquidity of the stool (R Santra, Calcutta School of Tropical Medicine, 2016).

Finally, it is important to note at this point that not everyone with hyperthyroidism will show hypermotility - some hyperthyroid patients show delayed gastric emptying (Pustorino et al. 2004). Indeed, constipation has been reported in patients with hyperthyroidism (Culp and Piziak 1986). Nonetheless, it is held that dysthyroidisms alter the motility of the gastrointestinal tract, and we do not completely understand how this works (Pustorino et al. 2004).