Here is a working example. I have tried to generate your scenario. Hope this will help.
<apex:page standardController="Account">
<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"></script>
<script type="text/javascript" language="javascript">
if(jQuery) {
jQuery.noConflict();
}
function changeColor(obj,tabname){
var selectedValue = obj.options[obj.selectedIndex].value;
var tabname = '[id$='+tabname+'_lbl]';
if(selectedValue == "inprogress"){
jQuery(tabname).css({"background-color":"red","color":"white","background-image":"none"});
}
if(selectedValue == "completed"){
jQuery(tabname).css({"background-color":"green","color":"white","background-image":"none"});
}
}
</script>
<apex:tabPanel switchType="client" selectedTab="name2" id="theTabPanel" activeTabClass="activeTab">
<apex:tab label="One" name="name1" id="tabOne">
<apex:form id="form1">
<apex:selectList id="selected_list" required="false" size="1" onchange="changeColor(this,'tabOne');">
<apex:selectOption itemvalue="None" itemLabel="--None--"/>
<apex:selectOption itemvalue="inprogress" itemLabel="In Progress"/>
<apex:selectOption itemvalue="completed" itemLabel="Completed"/>
</apex:selectList>
</apex:form>
</apex:tab>
<apex:tab label="Two" name="name2" id="tabTwo">
<apex:form id="form2">
<apex:selectList id="selected_list" required="false" size="1" onchange="changeColor(this,'tabTwo');">
<apex:selectOption itemvalue="None" itemLabel="--None--"/>
<apex:selectOption itemvalue="inprogress" itemLabel="In Progress"/>
<apex:selectOption itemvalue="completed" itemLabel="Completed"/>
</apex:selectList>
</apex:form>
</apex:tab>
</apex:tabPanel>
</apex:page>