user14790: My setup is as follows:

Arduino IDE 1.6.11 
Arduino Pro Mini as ISP 
ATTiny85 1602A LCD with I2C board

The libraries I use are:

TinyWireM (source: https://github.com/adafruit/TinyWireM)
LiquidCrystal_I2C (modified for ATTiny) (source: http://playground.arduino.cc/Code/USIi2c)

The Sketch I use to test with:
#include <TinyWireM.h>                 
#include <LiquidCrystal_I2C.h>         

LiquidCrystal_I2C lcd(0x3F,16,2);  // set address & 16 chars / 2 lines

void setup()
{

  TinyWireM.begin();                    // initialize I2C lib
  lcd.init();                           // initialize the lcd 
  lcd.backlight(); 
  lcd.clear();  // Print a message to the LCD.
}

void loop()
{
  lcd.setCursor(0, 0);
  lcd.print("Hello World on Attiny85");
  delay(2000);
}

The address is 0x3F, I found this using the scanner sketch, I also tested the LCD using the Arduino Pro Mini directly, it works fine.
The result of this setup is a single line of black blocks. My guess is that the initialization failed in some way. I have tried some things to make it work:

Pull-ups on SDA and SCL
Isolated from Arduino with regulated power supply
Setting contract with the pot meter on the back
reinstall of PC
reinstall of Arduino IDE
1MHz, 8Mhz, TinyCore default and High-Low Tech
Replaced FTDI
Replaced Tiny85
Replaced LCD
Replaced Arduino Pro Mini

Some pictures:



[EDIT]
Final sketch:
#include <TinyWireM.h>                  // I2C Master lib for ATTinys which use USI
// #include <LiquidCrystal_I2C.h>
#include "LiquidCrystal_attiny.h"
LiquidCrystal_I2C lcd(0x27,16,2);

void setup() {

  // Now set up the LCD
  //lcd.begin(16,2);               // initialize the lcd
  pinMode(4, OUTPUT);
  lcd.init();
  lcd.backlight();
  lcd.home ();                   // go home
  lcd.print("Weatherstation");
  lcd.setCursor ( 0, 1 ); // go to position
  lcd.print("BMP180");
  delay(2000);
}
void loop() {
  digitalWrite(4, HIGH);
  delay(100);
  digitalWrite(4, LOW);
  delay(100);
}


user14790: After hours of Googling, I found a blog post by Dimitris Platis where he describes using an ATTiny85 with I2C LCD. I emailed the guy and he replied with a link to his LCD I2C ATTiny library on GitHub. Using this library with the TinyWireM and the right core, it instantly worked.
My final setup is with a 4.7K resistor on SDA and SCL, I will add the final sketch in the original post under [EDIT]. This adventure took me two weeks so I hope it helps someone else some day.
LCD library used: https://github.com/platisd/ATtiny85-LCD-library
TinyWireM used: https://github.com/adafruit/TinyWireM
Core used: https://github.com/vprimachenko/ArduinoTiny

14790: <p>My setup is as follows:</p>

<ul>
<li>Arduino IDE 1.6.11 </li>
<li>Arduino Pro Mini as ISP </li>
<li>ATTiny85 1602A LCD with I2C board</li>
</ul>

<p>The libraries I use are:</p>

<ul>
<li>TinyWireM (source: <a href="https://github.com/adafruit/TinyWireM" rel="noreferrer">https://github.com/adafruit/TinyWireM</a>)</li>
<li>LiquidCrystal_I2C (modified for ATTiny) (source: <a href="http://playground.arduino.cc/Code/USIi2c" rel="noreferrer">http://playground.arduino.cc/Code/USIi2c</a>)</li>
</ul>

<p>The Sketch I use to test with:</p>

<pre><code>#include &lt;TinyWireM.h&gt;                 
#include &lt;LiquidCrystal_I2C.h&gt;         

LiquidCrystal_I2C lcd(0x3F,16,2);  // set address &amp; 16 chars / 2 lines

void setup()
{

  TinyWireM.begin();                    // initialize I2C lib
  lcd.init();                           // initialize the lcd 
  lcd.backlight(); 
  lcd.clear();  // Print a message to the LCD.
}

void loop()
{
  lcd.setCursor(0, 0);
  lcd.print("Hello World on Attiny85");
  delay(2000);
}
</code></pre>

<p>The address is 0x3F, I found this using the scanner sketch, I also tested the LCD using the Arduino Pro Mini directly, it works fine.</p>

<p>The result of this setup is a single line of black blocks. My guess is that the initialization failed in some way. I have tried some things to make it work:</p>

<ul>
<li>Pull-ups on SDA and SCL</li>
<li>Isolated from Arduino with regulated power supply</li>
<li>Setting contract with the pot meter on the back</li>
<li>reinstall of PC</li>
<li>reinstall of Arduino IDE</li>
<li>1MHz, 8Mhz, TinyCore default and High-Low Tech</li>
<li>Replaced FTDI</li>
<li>Replaced Tiny85</li>
<li>Replaced LCD</li>
<li>Replaced Arduino Pro Mini</li>
</ul>

<p>Some pictures:</p>

<p><a href="https://i.stack.imgur.com/GZePC.jpg" rel="noreferrer"><img src="https://i.stack.imgur.com/GZePC.jpg" alt="back of the display"></a></p>

<p><a href="https://i.stack.imgur.com/hTfyz.jpg" rel="noreferrer"><img src="https://i.stack.imgur.com/hTfyz.jpg" alt="the blocks"></a></p>

<p><a href="https://i.stack.imgur.com/DysXQ.jpg" rel="noreferrer"><img src="https://i.stack.imgur.com/DysXQ.jpg" alt="connections"></a></p>

<p>[EDIT]</p>

<p><strong>Final sketch:</strong></p>

<pre><code>#include &lt;TinyWireM.h&gt;                  // I2C Master lib for ATTinys which use USI
// #include &lt;LiquidCrystal_I2C.h&gt;
#include "LiquidCrystal_attiny.h"
LiquidCrystal_I2C lcd(0x27,16,2);

void setup() {

  // Now set up the LCD
  //lcd.begin(16,2);               // initialize the lcd
  pinMode(4, OUTPUT);
  lcd.init();
  lcd.backlight();
  lcd.home ();                   // go home
  lcd.print("Weatherstation");
  lcd.setCursor ( 0, 1 ); // go to position
  lcd.print("BMP180");
  delay(2000);
}
void loop() {
  digitalWrite(4, HIGH);
  delay(100);
  digitalWrite(4, LOW);
  delay(100);
}
</code></pre>
 14790: <p>After hours of Googling, I found <a href="https://platis.solutions/blog/2015/03/22/diy-interactive-name-tag/" rel="nofollow">a blog post</a> by Dimitris Platis where he describes using an ATTiny85 with I2C LCD. I emailed the guy and he replied with a link to his LCD I2C ATTiny library on GitHub. Using this library with the TinyWireM and the right core, it instantly worked.</p>

<p>My final setup is with a 4.7K resistor on SDA and SCL, I will add the final sketch in the original post under [EDIT]. This adventure took me two weeks so I hope it helps someone else some day.</p>

<p>LCD library used: <a href="https://github.com/platisd/ATtiny85-LCD-library" rel="nofollow">https://github.com/platisd/ATtiny85-LCD-library</a><br>
TinyWireM used: <a href="https://github.com/adafruit/TinyWireM" rel="nofollow">https://github.com/adafruit/TinyWireM</a><br>
Core used: <a href="https://github.com/vprimachenko/ArduinoTiny" rel="nofollow">https://github.com/vprimachenko/ArduinoTiny</a></p>
