For the sake of tracking, please file one issue per problem. A problem is a combination of:
- a specific device
- a specific monitor
- a specific cable/adapter

for all those three, please specify the exact model that is used so that we can reproduce locally.

 marcheu@ - As requested, I've split these into different issues (and added some additional notes below).  I didn't bother in some cases, as the original comments didn't seem to contain enough actionable info.  In the cases that I did file separately, I included all the info I had.  As noted before, these were all comments left in a bug that was more about the content of the message than its source(s).  For any of these new issues, feel free to "Won't Fix" them if you don't think they contain enough info.

More fine-grained triage of these issues, including "DEVICE / MONITOR / CABLE" summary where available.

1. ASUS Chromebox / Samsung 1080P screen (T27B300) / HDMI
"I don't know why it showed this message ("Dear Monitor"), but the resolution looks ok, xrandr detects the HDMI connection and the resolution without any problem including overscan (which also shows on the Chrome screen settings)."
Filed as Issue 640004

2. ASUS Chromebox / Medion MD 20999 / HDMI
Filed as Issue 640005

3. Samsung Chromebox i5 I/O edition / HP2711x / ???
"Same issue with Samsung Chromebox i5 I/O edition on my dual monitor setup. I run a HP2711x on the left and a Samsung S27B550 on the right and have for a well over a year and recently I have been getting the message about the HP model and it will occasionally black out and come back. Tried swapping ports and the HP is still having issues. The HP works fine when I connect to external port on OSX from my Macbook so I am ruling out faulty monitor, cable and port."
Filed as Issue 640006

4. Samsung 303C / Westinghouse SK-26H730S TV / HDMI
"There is a current problem on chromebook central where a TV connected by cable but TV is unpowered which is preventing the chromebook from booting. Remove hdmi cable and it boots OK.
 https://groups.google.com/a/googleproductforums.com/d/msgid/chromebook-central/04b28194-2384-44e1-8a87-3d0332027a05%40googleproductforums.com."
Filed as Issue 640007

5. "I just received this on my Acer C720. I don't have an external monitor."
No external monitor, see (*) below.

6. ASUS Chromebox / Toshiba TV / HDMI
 "It's a funny error message considering my setup. I have the ASUS Chromebox connected to a Toshiba TV via HDMI with a resolution of 1280x720 with no problems.  It's only when I power off the TV and then come back later do I see this new message: "Dear Monitor, it's not working out between us. (No supported resolution found.)"  Of course, as the screen lights back up, everything works normally."
Filed as Issue 640008

7. "Seeing this on one of my monitors as well on the Pixel 2. Identical monitors, both work, both HDMI cables work; verified with another (Windows) laptop."
Not enough info, Won't Fix.

8. "I have a recently purchased Asus C300EDU and am receiving this message. I have never attached an external monitor to it. The system is in developer mode, however, and I expect that it came up when switching between the ChromeOS desktop and Ubuntu Unity through Crouton."
No external monitor, see (*) below.

9. "I have recently started receiving this message on my Samsung Chromebook, as well. Also experiencing momentary screen blackout. When this message comes up, I lose the ability to use Google search engine-it just keeps "thinking" and eventually times out. I can type in the bing url and search through bing fine-but nothing (search bar, favorites, etc.) on the Google home page works. Only solution has been to restart and sometimes that doesn't work and I have to powerwash. I am NOT using an external monitor, never have on this device."
No external monitor, see (*) below.

(*) Apparently this sometimes appears when there is no external monitor (e.g.,Issue 444911, Issue 468134,  Issue 568176 ).  None of the reports here have enough info to reproduce them, so I'm just going to call them "Won't Fix".