TL;DR: Reading an email "with your eyes" is not the problem, since reading is not even a form of data processing. It's all the rest that might become a problem: what you do after reading it, what your systems (or third-party systems) do automatically with your emails, etc.
It all depends on what you are going to do with those emails, or even what you are doing without even realizing it (like automatic backups, scanning, processing, archiving, etc). You don't always need explicit consent to deal with personal data. For example you could process personal data on the basis of "legitimate interests", without any consent.
Let's suppose you only use the personal data in your emails to answer them and maybe do the job or task the sender requests. For example, somebody might send you an email saying they are interested in something you do, and asking you to call them back at a phone number. What are you going to do? Of course you can use that personal data to answer the email and make a phone call, and you can keep a backup of that data for some time (but not forever). All this is legitimate processing of personal data, it is expected and understood by the users, and doesn't need any consent.
Now let's suppose you use the personal data in your emails build a list of interests of the users, and send that list to someone else that will use it for commercial purposes. This would not be legitimate without an explicit consent. This means that you could not do this automatically for every email you receive, it would be illegal. You would have to check if the sender had already given you explicit consent for this kind of processing. It is ok to send them an email to ask for their consent. What you cannot do is to start processing their personal data for the above-mentioned commercial purposes without their explicit consent.
So "reading" an email is not a problem at all. The problem is what you are actually doing with the personal data after you read the email, or what you (or someone else, for example Google in Gmail?) might already be doing automatically even before you read it.
For more detailed information on this I think you need to look for "lawful basis for processing" or something like that in the GDPR law (Article 6). Also look for definitions and notes on "legitimate interests". An interesting quote on this, from the GDPR law (emphasis added by me):
The legitimate interests of a controller, including those of a controller to which the personal data may be disclosed, or of a third party, may provide a legal basis for processing, provided that the interests or the fundamental rights and freedoms of the data subject are not overriding, taking into consideration the reasonable expectations of data subjects based on their relationship with the controller. Such legitimate interest could exist for example where there is a relevant and appropriate relationship between the data subject and the controller in situations such as where the data subject is a client or in the service of the controller. At any rate the existence of a legitimate interest would need careful assessment including whether a data subject can reasonably expect at the time and in the context of the collection of the personal data that processing for that purpose may take place. The interests and fundamental rights of the data subject could in particular override the interest of the data controller where personal data are processed in circumstances where data subjects do not reasonably expect further processing.
