Is it dangerous to continue to remove a slightly punctured phone battery?
... had to remove the battery in the process.
  While doing so, I saw a tiny spark and felt a weird smell. I can't see any damage on the battery, but obviously it is no longer safe to use, and I am going to have to replace it, and dispose of the old battery properly. 


Given the above statements, your aims are removal of the battery without damage to
- yourself
- the phone

in that order.

Nothing is certain. You will probably be able to discharge the battery safely, but this is not 100% certain. 

The phone's location during the removal process should be somewhere where a battery fire will not matter.

Discharging the battery fully, either using the phone or by other means, before attempting removal is safer than not doing so.
If you can turn the phone on then, as long as it is functioning normally, it will discharge the battery over the "standby period" or less.
If you can access the battery wires or terminals then you can discharge the battery using a resistor. This is a surer method but harder to do if inexperienced. 

If it has a "flashlight" then this will usefully reduce discharge time.
The battery will still retain some energy when the phone decides that it is 'flat' but will be much less liable to cause damage.
(A 1500 mAh x 3.6V battery with 1% charge has about 150 Watt seconds of energy remaining. The same energy as is required to raise a 1 kg weight 15 metres - not to be treated too lightly). 


  ... is it safe to use a heat gun to loosen up the glue?


Maybe, but not wise. Do not do unless essential AND you are willing to have the phone destroyed.
Even using the absolute minimum of temperature, the odds are the rise needed to be useful is above what the phone itself will withstand. 


  Also; could it be dangerous to leave it on the kitchen table over night? 


Yes - potentially so. As above, leave phone somewhere that a fire does not matter (except to the phone). 


  Should I wear gloves when doing this now?


May help but adds fumble-fingeredness.
Probably good enough to use care, prompt action, eye protection, sensible clothes. LiIon batteries usually do not "explode" - they transition to a flaming heap - sometimes over a minute or so - or occasionally in seconds. I have NEVER had this happen but my all-care-no-responsibility comment is that I'd act as above with the awareness of the unlikely but possible worst case results. 



Note the article cited by James Large in comments
Toxic fluoride gas emissions from lithium-ion battery fires. I'd be suitably wary of the fumes - but no more so than with fumes from many other electronic and chemical systems. ie aim to breath none, make it extremely unlikely that you'll get more than a whiff worst case, use of a fan or ventilator is liable to help. Note their notes on using water mist for extinguishing such fires. Water is not recommended as it reportedly may increase Lithium combustion rate and also the rate of HF production (but not the total HF produced).