I have modified the code..

trigger BeforeCaseInsert on Case (before insert,before update)
{
List<String> RecordTypeNameList = {'JIRA','Prod Specialist','BenchPress','Support'};

List<RecordType> RecordTypeResult = [SELECT id,Name FROM RecordType WHERE name in: RecordTypeNameList];

Map<String,Id> RecordTypeNameIDMap = new Map<String,Id>();

for(RecordType rt : RecordTypeResult)
{
RecordTypeNameIDMap.put(rt.Name,rt.id);
}

/* private final Id recordTypeId = [SELECT id FROM RecordType WHERE name =: 'JIRA'].id;
private final Id recordProdSquadID = [SELECT id FROM RecordType WHERE name =: 'Prod Specialist'].id;
private final Id recordBPID = [SELECT id FROM RecordType WHERE name =: 'BenchPress'].id;
private final Id recordSupport = [SELECT id FROM RecordType WHERE name =: 'Support'].id; */
private List<Id> lstAccount = new List<Id>();
private Map<Id, Account> accMap = new Map <Id,Account>();
private List <Sites__c> sitesF;

// Get a list of Accounts
for (Case c : trigger.new){
lstAccount.add(c.accountId);
}

// Create a Map with AccountId, Account to reference later in the code
for (Account a : [Select id,publication_Manager2__c from Account where id in: lstAccount]){
accMap.put(a.id,a);
}

sitesF = new List<Sites__c>([SELECT id, Name, Feedback__c, Account__c, Account__r.Publication_Manager2__c
FROM Sites__c LIMIT 50000]);

for (Case c : trigger.new) {
for(Sites__c s : sitesF) {
if (s.Feedback__c != null
&& c.Description != null
&& c.Description != ''
&& c.recordTypeID == RecordTypeNameIDMap.get('Support')
&& c.Description.contains(s.Feedback__c)) {
c.sites__c = s.id;
c.accountId = s.account__c;
accMap.put(s.account__c,s.Account__r);
break;
}
}
// This is a second bit of code
// if (c.recordTypeId == RecordTypeNameIDMap.get('JIRA')
// && accMap.get(c.accountId)!= null
// && accMap.get(c.accountId).Publication_Manager2__c != null) {
// c.OwnerId = accMap.get(c.accountId).publication_Manager2__c;
// }

// Paul code
if (c.recordTypeId == RecordTypeNameIDMap.get('Prod Specialist')
&& c.JCode__c != null
&& c.JCode__c != ''
) {
for(Sites__c s : sitesF) {
if (c.JCode__c == s.Name){
c.accountId = s.account__c;
c.Sites__c = s.id;
accMap.put(s.account__c,s.Account__r);
break;
}
}

}
if (c.recordTypeId == RecordTypeNameIDMap.get('BenchPress'))
{
for(Sites__c s : sitesF) {

if (c.Journal_Code__c == s.id){
c.Sites__c = s.id;
c.accountId = s.account__c;

accMap.put(s.account__c,s.Account__r);
break;
}
}
}

}

Hope this solves your problem... I think the error may be too many SOQL statements 101 when you perform bulk trigger...

If this code doesn't solves your problem, then provide me details about the issue which help me to fix the issue..