To answer the original two questions:
The plants can get puffy/fat during times of heat and humidity but it can vary a lot between courses and management - with some experiencing more than others. So it can be a factor. Usually it means that the turf will scalp when mowed, which will add extra stress at at a point of the year where there is a lot of stress anyways. If it happens it is a good indicator that it might be time to double down and get them through the summer.
There are tools that can help dehumidify greens like fans as mentioned and sub air type systems that blow air though the drainage and into the rootzones. The biggest tool is to try and get as much air into the rootzone as possible, via various non distrupive aerations like needletining. Topdressing can help also with the puffyness but adding sand in the heat of the summer may add more stress so is not always the correct solution.
To answer the thread title is a lot more involving. So much depends on the weather and the growing environment. For example in the MidAtlantic where the day time temperatures and humidity can get really high then then a cool season grass is just going to try to go dormant to survive. If there is a cooler summer where the temperatures stay in a range where cool season turf if happy then, in simple terms, they could be dried out and survive July and August. The location on a course can make a huge difference also as shaded pocketed greens may have issues whereas nearby greens that are in the open may be fine.
Generally greens are watered to a minimum (like Justin mentioned) as too much water causes more issues, but drying them beyond the wilt point will cause longer term damage and, if death occurs, an expended period of healing in the fall when they normally can be pushed without worry. Remember it's always easier to add water then remove it.
There are a lot of tools available to monitor water usage and provide a guideline of when/how much water is needed. However - from experience - there is still some art involved as these tools don't always take all the factors into consideration so blindly following them can cause issues.
Here are a few links that explain in more detail for those who are interested.
Blog post on syringing with pics showing the difference in temperature
http://ledgerockgcmaintenance.blogspot.com/2014/07/when-watering-isnt-watering.htmlSome info on fans
http://ledgerockgcmaintenance.blogspot.com/2013/05/turf-fans.htmlThe end of this blog entry shows the difference fans make to rootzone temperature
http://ledgerockgcmaintenance.blogspot.com/2013/07/the-end-of-heat-wave-number-3.htmlLastly here are two USGA green section articles that may help
http://gsrpdf.lib.msu.edu/ticpdf.py?file=/2010s/2013/2013-06-28.pdfhttp://gsrpdf.lib.msu.edu/ticpdf.py?file=/2000s/2006/060313.pdf