This is a question that arose in the Microsoft Tech Community (take a look at this thread) that as far as I know, it does not have a clear answer since there is not official Microsoft documentation about the side effects of disabling Delve in an Office 365 tenant. However, based on the knowledge we have about the use of the Office Graph across Office 365 we can think that the following side effects could happen when disabling Delve:
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Office 365 Home Page:
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Recent Online Documents.
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Recent OneDrive Folders.
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SharePont Online:
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SharePoint landing page: Suggested sites.
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Site Content Page: Trending Content.
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OneDrive for Business:
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Discover feature.
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Copy files from ODFB to SPO Sites (Frequent Sites).
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People page:
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Suggested People.
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People for Follow-up
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New contact Card that it’s used everywhere in Office 365.
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Office 365 Video: Trending Videos.
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Other possible side effects: Clutter in Exchange Online, SharePoint and Office 365 Groups Mobile Apps, Discover feature in Office 365 Groups, Security and Compliance Center, etc.