Dynamically resize VisualForce section in page layout

Yes it is possible. See the Update - Success at the bottom:

Old Answer - close but not so much

Resizing the VF Page within a Page Layout is not possible though there is an option on the Page Layout that allows scrollbars for the embedded content.

Another option is to (1) make the "embedded" Visualforce Page a Visualforce Component, and (2) write a VF Page similar to:

<apex:page standardController="Your_Object__c">
    <apex:detail subject="{!Your_Object__c.Id}" relatedList="true"/>
    <c:yourComponent recordId="{!Your_Object__c.Id}" />
</apex:page>

Then override the View Standard Button or Link on the Object page with this VF Page.

This option does have some minor downsides in that developers will have to go into the Object from the Setup to change the Page Layout rather than the using the "Edit Layout" link on the top-right. Also Developers will lose the Dev Tools Menu. Again, these are minor issues for Developers, but this could add the functionality you are looking for. It's worth a try.

---

Update - Success

VF Pages embedded in a Page Layout come over as iframes, so the solution is to write a VF Page using only <apex:detail .../> to bring over the Page Layout and then have JavaScript (via jQuery) resize all the iframes.

Example

<apex:page standardController="Custom_Object__c">
    <apex:detail subject="{!Custom_Object__c.Id}" relatedList="true"/>  
    <apex:includeScript value="//ajax.googleapis.com/ajax/libs/jquery/1.11.1/jquery.min.js"/>
    <script>
        var j$ = jQuery.noConflict();
        j$("iframe").each(function(){
            j$(this).load(function() {
                j$(this).height( j$(this).contents().find("body").height() );
            });
        });
    </script>
</apex:page>

Some more ideas on this include using HTML 5 postMessage() and cross-domain messaging for iframes. I've written up a blog post on how this can be used to not only do dynamic iframe resizing, but also implement basically any collaboration between the parent page and embedded VF that your business requires. HTH

http://salesforcehacks.blogspot.com/2014/12/resizing-embedded-vf.html

Yes it is possible.

@AmatorVitae's answer will certainly work if you can set up your visualforce page to use apex:details.

If you need more than apex:detail

To resize a visualforce page with a more complex structure, you can do so with some cross domain scripting.

In my answer here, I go into detail about how to accomplish this.

To summarize

You need to

1. Add a custom link section to your sidebar that executes some javascript stored in a static resource.
2. Add a static resource that is set up to receive a message from another domain and then use the message's value as the new height of the iframe.
3. Add some script to your visualforce page that will send a message to the parent domain with the value of the content's height.