Can reaction of gastric acid with swallowed things be dangerously exothermic?
For an average healthy adult person maximum concentration of hydrochloric acid 
in gastric juice is $C = \pu{0.16 M}$, and on average there is $V = \pu{2 L}$ of gastric acid in the stomach, assuming it's daily secretion of roughly $\pu{1.5 L}$ and some leftovers. Considering that the acid hasn't been used for the purpose yet, and the following reaction

$$\ce{Al + 3HCl -&gt; AlCl3 +1.5H2},\tag{1}$$

one can make an assumption that the mass of aluminium metal that can possibly be digested is

$$m_\ce{Al} = \frac{1}{3} CVM_\ce{Al} = \frac{1}{3} \times \pu{0.16 mol L^{-1}} \times \pu{2 L} \times \pu{26.98 g mol^{-1}} = \pu{2.85 g}.$$  

In reality the outcome depends on the health conditions and biometrics of the given individual. People suffering from acid deficiency won't be able to digest even this amount. Larger or overweight specimen can have larger/stretched stomach, so the amount of acid and its pH may vary quite a lot. 

On the other hand, having certain decease such as gastric ulce may amplify the effect of digestive system reacting to the produced $\ce{AlCl3}$ and heat. Also, aluminium(III) chloride is considered to be a neurotoxin. Acute oral
toxicity ($\ce{LD_{50}}$, mouse) of $\ce{AlCl3}$ is $\pu{3805 mg/kg}$. Reaction (1) produces approx. $\pu{14 g}$ of $\ce{AlCl3}$, which is not all that bad. 

As @MaxW noticed in the comments, swallowing any metal is potentially dangerous and may or may not cause unwanted effects.