Do groups require Office 365 license?

Power BI groups do not require an Office 365 subscription! If you do have an Office 365 subscription, you get to take advantage of extra features like connectiong to a group's OneDrive account, but without a subscription, you can still create and manage groups and connect to the other data sources.

For some options for the Office 365 Groups you need an exchange license, strangly enough.

If you want to turn of the Welcome Email for the Office 365 Group for example you need an exchange license.

Powershell commands to do this:

$groupName = "GroupName"
$UserCredential = Get-Credential
$Session = New-PSSession -ConfigurationName Microsoft.Exchange -ConnectionUri https://outlook.office365.com/powershell-liveid/ -Credential $UserCredential -Authentication Basic -AllowRedirection
Import-PSSession $Session
Set-UnifiedGroup $groupName -UnifiedGroupWelcomeMessageEnabled:$false
Remove-PSSession $Session

Error you get:

The following error occurred during validation in agent 'Substrate Only Agent': 'Organization "org.onmicrosoft.com" i
s not licensed for Exchange email functionality. Cmdlet usage is restricted.'
    + CategoryInfo          : NotSpecified: (:) [Set-UnifiedGroup], ProvisioningValidationException
    + FullyQualifiedErrorId : [Server=VE1P189MB1149,RequestId=b45e9174-9de6-45f4-994a-e953d8b4ab64,TimeStamp=19-3-2020
    12:12:35] [FailureCategory=Cmdlet-ProvisioningValidationException] F7111EC1,Microsoft.Exchange.Management.Recipie
  ntTasks.SetUnifiedGroup
    + PSComputerName        : outlook.office365.com

So for adding, deleting an basic changes you do not need a license but for more advanged stuff you need an Exchange license (or office which contains exchange i believe).