Does it highly affect the stand-by time to leave wi-fi open all the time on my phone?
Yes, I would imagine leaving WiFi on all the time would drain the battery faster. Even if you're not actively connected to an Access Point (AP), the phone is still constantly scanning all the APs in the vicinity of where you are and that takes energy.
The general rule is to turn off what you are not using.
This has been my experience with HTC's Evo:
WiFi uses less power compared to 3G when actively downloading/uploading large amounts of data. Even when doing a lot of browsing. However, unlike 3G, the WiFi radio never goes to sleep. So when you aren't actively downloading data, it's best to turn it off. This is unlike GPS and Bluetooth where if they aren't being used, they don't use much power at all.
Turning off anything that isn't in use will generally increase battery life. Also remember that the biggest drain on the battery is generally the display. Setting your brightness to the lowest level (or at least auto) can do a lot to save your battery life. Also, if you typically connect to WiFi at home, but don't have access at your work, there are apps that can automatically turn off the WiFi radio (among other things) in the morning, then turn it back on in the late afternoon/early evening. I use Advanced Volume Mode Scheduler to do this.
Wifi uses significantly less power then 2G or 3G and in fact uses less while connected and inactive than when it is searching for an AP. So I leave it on all the time and my battery lasts longer.
As JamesRyan pointed out, leaving WiFi connected and inactive uses less battery than if it has to keep searching for an AP.
I use a HTC Desire and initially had my WiFi set to sleep after 15min, but then realised that every time I unlocked my phone it would wake the WiFi, and in turn it would start searching for an AP. Afterwards, I set WiFi to never turn off - this actually resulted in more battery power left at the end of the day (even with the occasional sync of my accounts - email, twitter etc).
By my experience it seems that if WiFi is connected to a network then leaving it on will improve standby time as all the periodic data access that an Android phone does will go through WiFi. And accessing data over WiFi is more battery efficient than 2G/3G/LTE.
On the other side, if WiFi is not connected to a network then it will constantly search for one and decrease the standby time.
So what I try to do is to leave WiFi on whenever I think I will be around a known WiFi AP for some time but I leave it off the rest of the time. The drawback is that it forces me to change the setting at least a couple times per day.
Yes, it decreases standby time because WIFI is one of the more significant things draining the battery. As long as WIFI is enabled, your phone is constantly scanning for new networks and "listening" on TCP ports.
Additionally it will be sending data from applications, requesting updates, etc.; and if you have the Play Store set to automatically update apps then it will also be downloading and installing apps, which also significantly drains the battery.