Aluminium, heat and biological safety
The use of aluminum in the construction of your ashtray as you've described will not result in the ingestion/inhalation of any form of aluminum in any significant way.  

All aluminum that has been exposed to air is covered in a chemically and physically stable oxide. This aluminum oxide has an even lower vapor pressure than aluminum metal, such that the tip of the cigarette will actually be more than $\pu{2000^oC}$ below it's boiling point. By physically stable, I mean that aluminum oxide adheres strongly to the bulk aluminum and will not flake off as in the case of many other metal oxides.  

Although your question was about potential safety issues, I will make one comment regarding your discussion of the design. You suggested that heat transfer into the aluminum might play a role in extinguishing the cigarette, which is not the case. So long as there is sufficient oxygen available for the combustion, the cigarette will continue to burn.