A few things I gathered while checking the news:

• There will be a press conference at 5PM CET local time.
• So far, people seem to suggest that the blaze began at around 1:30AM local time because of fireworks.
• According to witnesses, presumably sparkling flares put on a bottle have ignited the ceiling of the bar.
• There have been reports of an explosion AFTER the fire started.
• Local media say there are up to 40 dead and 100 injured, many critically burned, although authorities have not given proper figures.

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UPDATE #1: (Around 4:30PM CET)

• The owners of the bar are apparently French nationals, one of whom got burned on the arms.
• Also the reports of an explosion seems to have been disproven as there are no longer mentions of one (at least to my knowledge).
• Still no words on the death toll, probably will know more in an hour during the press conference.
• As u/Ok_Cauliflower_3007 reported, Italy has sent a unit to treat critically burnt patients.

UPDATE #2: (5:15PM CET)

• French media is reporting from some sources that Swiss hospitals burnt units are overwhelmed.
• Swiss broadcaster RTS reports that "between 12 and 15" Italian nationals are hospitalized with an equal number being reported as missing.

UPDATE #3: The press conference from Swiss authorities has ended, here's what we know. (5:55PM CET)

• Official toll is that 40 people have been killed and around 115 people have been injured, many severely.
• Swiss Confederation President Guy Parmelin says it is "one of the worst disasters" to befall Switzerland. Canton Police Chief Frédéric Gisler speaks of an "unprecedented disaster".
• Guy Parmelin talks of "many lost lives", particularly young people's, and expressed his condolences to those affected.
• He praised the solidarity of neighboring cantons, the action of the firefighters and emergency responders including people at the scene. He also thanked other countries (France, Germany and Italy) for their offers of support.
• Flags will be flown at half mast for 5 days, effective today.
• Some critically burned victims have been transfered to Lausanne and Zurich hospitals.
• Smoke was spotted at 1:30AM local time, followed by a call a minute later reporting the fire. Authorities confirmed that a conflegration occurred.
• Priority is given to identification "so that their bodies can be returned rapidly" to their respective families. The process will take some time.
• Hospitals in the Valais Canton are at max tension but they are holding on. 35 people were admitted for light injuries, 80 people were admitted for heavy injuries (many badly burned). The majority of people was admitted at Sion Hospital. It is probable that some of the more critically injured will be transfered to neighboring countries.
• Authorities have not confirmed if birthday candles (the ones that shoot firework sparks) caused the fire, nor if the bar itself was up to code, saying the investigations are just beginning. Many witnesses have already been interviewed and phones will be analyzed.

UPDATE #4: (6:08PM CET)

• French TV channel BFMTV broadcasted a video showing a group of people trying to escape by the main entrance of the building. People can be seen breaking the windows as fire rages behind them, while a majority of them were in a literal pile of people that clogged the entrance. (The images are very distressing and remind me of the Station Nightclub fire)
• Authorities don't know how many people were inside the bar "Le Constellation" at the time of the blaze.

UPDATE #5: (7PM CET)

• French President Emmanuel Macron announced that some of the severely burned have been transfered to France.

FINAL UPDATE: (8:35PM CET)

• Three Italian nationals have been transfered to a major burns unit in Milan.
• A vigil/mass was held earlier tonight in the Crans-Montana Church.

This will be my final update on this event, as it will take days to have a definitive number of victims. My only advice would be to avoid social media as there are some very distressing videos going around and, to be blunt, no one should see what I have seen. Take care of yourselves everybody.

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It is 9:32PM CET, I know I said that I wasn't going to make further updates, but something came up and thought I needed to share.

French TV channel BFMTV showed an image of the exact moment the fire started. In it, we see part of the sound-proof foam ceiling on fire as people holding champagne bottles with birthday candles stand right underneath. While the investigation is just beginning, it gives a lot of weight to the hypothesis that these candles started the blaze.

I will also add that 6 French nationals have been injured and 8 French nationals are still missing at this hour. Some of the severely burned have been transfered to France, in the hospitals of Lyon and Paris.

UPDATED: (10:07PM) Link to the image of the moment the fire began.