Let us evaluate the validity of all the statements:

1. Alkyl amines are stronger bases than aryl amines (True): In aryl amines (like aniline), the lone pair of electrons on the nitrogen atom is delocalised over the benzene ring via resonance, making it less available for protonation. In contrast, alkyl groups exert an electron-donating inductive effect (+I effect), which increases the electron density on the nitrogen atom in alkyl amines.
2. Alkyl amines react with nitrous acid to produce alcohols (True): Primary aliphatic amines react with nitrous acid ($HNO_2$) to form highly unstable aliphatic diazonium salts, which rapidly decompose to yield primary alcohols and nitrogen gas.
3. Aryl amines react with nitrous acid to produce phenols (False): Primary aromatic amines react with cold nitrous acid (at 273-278 K) to undergo diazotisation, forming relatively stable arenediazonium salts. Phenols are only produced if these diazonium salts are subsequently boiled or heated with water.
4. Alkyl amines are stronger bases than ammonia (True): The +I effect of alkyl groups in alkyl amines increases the electron density on the nitrogen atom compared to ammonia, making them stronger bases.

Therefore, the false statement is that aryl amines react with nitrous acid to produce phenols.