In order to sync updates from HubSpot records to Salesforce, the integration user needs to be able to access and edit the corresponding field or object in Salesforce.
A permissions error occurs when a Salesforce field or object is not visible or editable while logged in as the integration user. You can find the username used to authorize the integration (the integration user) by navigating in HubSpot to Account Menu > Integrations > Settings (under Salesforce).
To resolve this error, ask the integration user to log into Salesforce and confirm that they have view all and modify all access on the Object (Lead, Contact, Account, Opportunity, Campaign, Task, or the HubSpot Intelligence object) that is syncing. If you do not have access to the HubSpot Intelligence object, assign the HubSpot Integration Permission set to the integration user record.
Next, check the field accessibility to ensure that the integration user has permission to read/write on the field that is specified in the permissions error. If not, please check out this resource for instructions on setting field level security.
Take a look at the Salesforce Profile and Role used by the integration user. We recommend that the Role be outside of any existing hierarchical Permissions schemes. More information on Role hierarchies in Salesforce can be found here.
After re-setting the integration user's permissions, re-sync the contacts that displayed the error by navigating to the sync errors screen in your Salesforce settings.
For more information on fixing Salesforce sync errors, check out this article.