Not surprising. Using PS-generated code as a real web page is usually a fatal mistake.
If you want to know why this is happening, then we will need a link to the page (that means you will have to upload it somewhere), or as a much poorer alternative, paste the code for the page into a reply here.
First, your page is much too wide - at 1456px, it's more than 50% wider than the usual lowest common screen size (1024). I never make pages wider than 980px.
Anyhow, where are the 'tiny lines' you are talking about?
If I 'tap' it up in Safari Browser to 200% there are very fine lines showing the slices. It does not show up even at 400% in DW Live View.
Also, supposedly my header is 1224 wide. According to everything on this computer.
Mac OS X Mountain Lion with Adobe CS6 running.
<table width="1456" border="0">
That's too wide.
If you are going to use a table-based layout, then your table tags need to have cellpadding="0" and cellspacing="0" attributes added to tighten the cells up.
In Safari and OSX 10.8, I do not see any breakage in that table containing the header image at any text size setting. Why did you slice that image the way you did anyhow? Why not just use a single image?
There should be no specified widths or graphics in your code that are greater than 980px total. You have several.
Also, I'm still curious why you sliced the header image as you did? Why not just use a single image.
<table width="1456" border="0">
<td width="1450" height="92"><table width="1231" border="0">
<td width="1225" height="86">
I panicked when you said the width was 1456 or whatever because I didn't realize you were looking at the code. I thought you were seeing something on the actual interface that I wasn't seeing.
The outside table is pulled out to 1456 (or whatever, some are pulled out further and some less ... I just pulled it out to get it out of my way) so that I can work on the inside tables without the outside one getting in the way.
This website is under construction. I have two websites that are under construction at the moment. I am using the same basic layout for both.
Basically this is a magazine layout design and extremely flexible in presenting all types of possible material. These are websites that can be built up over time without having them become a confusing mess.
The header size was based on a number of books done by top websites designers, 1248 px wide with 910px being the basic area of focus. At the time it had to do with screen resolution and was before HD. That in turn seems to be based on the 17 X 11 tabloid format for magazine layouts. What builds in Illustrator at 17" wide ends up being a few pixels short in Photoshop at 1224 pixels.
The Black Forest website was my first experiment with the type of header and layout that I seem to find most useful when handling all kinds of mismatched material and transitional material. Transitional covering decades of artwork ... starting well before the advent ot the computer. I also was using GoLive! to build it. Before I had GoLIve! I was building simple sites writing all of my own html code and tables. It still works well considering I haven't touched it in 5 years. I will need to rebuild it at some point.
The sliced header .... I have experimented with that, too. The other two websites have a graphic image that covers the top part of the header and the menu bar. Vincent Charles is the only header that is strictly font at the moment. They may add a graphic image (custom vector) to it later. I did figure out that the reason the slice lines show is because I am using a light grey on a dark background. The backgound shows through. It is miniscule. I just thought something was wrong. I looked at my other two similar headers and realized the lines were there also, but I never saw them because the colors are darker and blend better.
To fit in the viewable area of the average user's display.
Google labs - Browser Size
Just because somebody has a 2300+px wide display doesn't mean they keep it maximized to anything near that. Also don't forget that many internet users are on mobiles & tablets now. Some smartphones are getting wider while Tablets are getting smaller.