Here's the final code, updated for deleting/updating/etc. No need to create a second trigger, just use the Trigger Context Variables. Works for Leads and Contacts, make sure to create the Open Activities field on both before creating the trigger. There's also a test method below, although you should probably add test methods for the ones you found: Deleting task, changing the task object.

trigger UpdateLeadOpenTasks on Task (after delete, after insert, after undelete,
after update) {

// Declare the variables

public set<Id> LeadIDs = new Set<Id>();
public list<Lead> LeadsToUpdate = new List<Lead>();
public set<Id> ContactIDs = new Set<Id>();
public list<Contact> ContactsToUpdate = new List<Contact>();

// Build the list of Leads and Contacts to update
if(Trigger.isInsert || Trigger.isUnDelete || Trigger.isUpdate){
	for(Task t: Trigger.new){
	if(string.valueOf(t.WhoId).startsWith('00Q'))
	LeadIDs.add(t.WhoId);
	if(string.valueOf(t.WhoId).startsWith('003'))
	ContactIDs.add(t.WhoId);
	}
}

if(Trigger.isDelete || Trigger.isUpdate){
	for(Task t: Trigger.old){
	if(string.valueOf(t.WhoId).startsWith('00Q'))
	LeadIDs.add(t.WhoId);
	if(string.valueOf(t.WhoId).startsWith('003'))
	ContactIDs.add(t.WhoId);
	}
}

// Update the Leads

if(LeadIDs.size()>0){
for(Lead l: [Select l.Id, l.Open_Tasks__c,
(Select Id From Tasks where IsClosed = False)
From Lead l where Id in :LeadIDs])
LeadsToUpdate.add(new Lead(Id=l.Id, Open_Tasks__c = l.Tasks.size()));
update LeadsToUpdate;
}

// Update the Contacts

if(ContactIDs.size()>0){
for(Contact c: [Select c.Id, c.Open_Tasks__c,
(Select Id From Tasks where IsClosed = False)
From Contact c where Id in :ContactIDs])
ContactsToUpdate.add(new Contact(Id=c.Id, Open_Tasks__c = c.Tasks.size()));
update ContactsToUpdate;
}


}

@isTest
private class TestOpenTaskUpdate {
static testMethod void LeadTest(){
// Create a Lead
Lead newLead = new Lead(LastName='Test', Company='ABC', Status='Targeted');
insert newLead;

// Create a Task
Task newTask = new Task(Subject='Open Task', Status='Not Started', WhoId=newLead.Id);
test.startTest();
insert newTask;
test.stopTest();

// Verify that the # Open Tasks is correct
newLead = [select Open_Tasks__c from Lead where Id=:newLead.Id ];
System.assertEquals(1,newLead.Open_Tasks__c);
}

static testMethod void ContactTest(){
// Create a Contact
Contact newContact = new Contact(LastName='Test');
insert newContact;

// Create a Task
Task newTask = new Task(Subject='Open Task', Status='Not Started', WhoId=newContact.Id);
test.startTest();
insert newTask;
test.stopTest();

// Verify that the # Open Tasks is correct
newContact = [select Open_Tasks__c from Contact where Id=:newContact.Id];
System.assertEquals(1,newContact.Open_Tasks__c);
}

static testMethod void CompletedTest(){
// Create a Lead
Lead newLead = new Lead(LastName='Test', Company='ABC', Status='Targeted');
insert newLead;

// Create a Completed Task
Task newTask = new Task(Subject='Open Task', Status='Completed', WhoId=newLead.Id);
test.startTest();
insert newTask;
test.stopTest();

// Verify that the # Open Tasks is empty
newLead = [select Open_Tasks__c from Lead where Id=:newLead.Id ];
System.assertEquals(0,newLead.Open_Tasks__c);
}

static testMethod void BatchTest(){
// Create 100 Leads
List<Lead> newLeads = new List<Lead>();
for (Integer i = 0; i < 200; i++) {
newLeads.add(new Lead(LastName='Test '+i, Company='ABC', Status='Targeted'));
}
insert newLeads;

// Create a task for each one
List<Task> newTasks = new List<Task>();
for(Lead l : newLeads){
newTasks.add(new Task(Subject='Open Task', Status='Completed', WhoId=l.Id));
}

// Insert the tasks
test.startTest();
insert newTasks;
test.stopTest();

// We could verify that the Open Tasks fields were updated correctly,
// but this is just testing the governor limits so it's not necessary
}
}

Oh, and the update is outside of the for loop. There are no brackets around the for loop, so it should only do the next line of code. You could put in brackets if you're worried.