Wenger, 67, was charged with verbally abusing and pushing fourth official Anthony Taylor after being sent off.
Wenger reacted angrily to Burnley being awarded a 93rd-minute penalty.
He misses Saturday's FA Cup tie against Southampton and Premier League games against Watford, Chelsea and Hull.
Burnley scored the penalty at Emirates Stadium on Sunday to level the scores at 1-1.
After being sent to the stands by referee Jon Moss, Wenger moved away from the pitch but stood at the tunnel entrance and refused to move as he tried to watch the remaining few minutes of Sunday's match.
As Taylor encouraged him to move away, Wenger was seen to push back against him.
Arsenal were then given a penalty of their own, which was converted by Alexis Sanchez.
Immediately after the match, Wenger apologised, saying: "I should have shut up - I apologise for not having done that.
"It was nothing malicious. I should have kept my control, even if it was in a hectic time."
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It will be the robot's first foray into healthcare after previous deployments in shopping centres, banks and train stations.
One hospital plans to deploy more robots over the next 10 years.
Some experts have questioned the usefulness of social robots such as Pepper.
Softbank, the company behind Pepper, and partner French robotics firm Aldebaran have seen huge success with the creation, with each batch of 1,000 selling out fast in Japan.
The hospital-based robots will be significantly more expensive than the $1,850 basic model with a $34,000 (Â£24,000) price tag.
This will include specially designed software from Belgian firm Zora Bots, as well as access to the helpdesk should issues arise when Pepper is on duty.
The 1.2m (4ft) tall robot can recognise the human voice in 20 languages and can detect whether it is talking to a man, woman or child.
In the CHR Citadelle hospital in Liege, Pepper will remain in the reception area but, at the AZ Damiaan hospital in Ostend, it will accompany visitors to the correct department.
The Liege hospital wants to become one of the world's most futuristic over the next 10 years.
"It's about architecture, future rooms and robots, the ones we already have and the ones we're going to get," said CHR Citadelle's communication director Nathalie Evrard.
Both hospitals already use the smaller Nao robots, also made by Softbank, in both their paediatric and geriatric wards.
The little robots assist in exercise sessions and help children overcome their fears of surgery, sometimes going to the theatre with them.
Some remain sceptical about how useful Pepper can be in both homes and commercial settings, with robotics expert Prof Noel Sharkey previously telling the BBC: "It is good at conveying human-like gestures but the way it detects human emotions might be over-hyped."
Arsenal manager Arsene Wenger has been given a four-match touchline ban and £25,000 fine after accepting a Football Association charge of misconduct for his behaviour in the win over Burnley.
[NEXT_CONCEPT]
Pepper, the humanoid robot programmed to "understand" human emotions, is to take a new job - as a receptionist in two Belgian hospitals.